In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.

After re-working the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics.

Some classical construction problems of geometry are impossible using compass and straightedge, but can be solved using origami.

In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first-and second-order equations, e. g., y

Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i. e., for any proposition P, the proposition " P or not P " is automatically true.

Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light.

The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed.

Phenomena that in classical mechanics are ascribed to the action of the force of gravity ( such as free-fall, orbital motion, and spacecraft trajectories ), correspond to inertial motion within a curved geometry of spacetime in general relativity ; there is no gravitational force deflecting objects from their natural, straight paths.

The term is also used as a collective term for the approach to classical, computational and relativistic geometry that makes heavy use of such algebras.

In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics.

The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect ; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

In general, separability is a technical hypothesis on a space which is quite useful and — among the classes of spaces studied in geometry and classical analysis — generally considered to be quite mild.

Many theorems from classical geometry hold true for this spherical geometry as well, but many do not ( see parallel postulate ).

In the classical geometry of space, a vector exhibits a certain behavior when it is acted upon by a rotation or reflected in a hyperplane.

In classical geometry, axioms are general statements while postulates are statements about geometrical objects.

In classical geometry, a proposition may be a construction that satisfies given requirements ; for example, Proposition 1 in Book I of Euclid's elements is the construction of an equilateral triangle.

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

Using this, it becomes relatively easy to answer such classical problems of geometry as

The design studies for precast concrete units or for the moulds for in situ shuttering, prompted by the need to obtain a large number of different forms from the combination of a very limited number of units contributed, in the 1980s, to the Taller ’ s affirmation of the validity of classical forms and geometry in contemporary architecture.

The classical velocity of an object is a tangent vector to its path.

The components of the classical velocity at a point p ( tangent to the curve ) are

Another, more classical, form of handrailing which is still in use is the tangent method.

In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangent space at each point.

* A classical covariant derivative or affine connection defines a connection on the tangent bundle of M, or more generally on any tensor bundle formed by taking tensor products of the tangent bundle with itself and its dual.

Both the tangent piano and fortepiano offered a variety of sound that was appealing to the changes in classical music, which featured more expressiveness and intensity than the harpsichord could offer.

Many kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including saddle node bifurcations, Hopf bifurcations, umbilic bifurcations, period doubling bifurcations, reconnection bifurcations, tangent bifurcations, and cusp bifurcations.

The open ceiling, with allegorical and classical figures thrown in masses against the sky: the closed frieze, formally divided into historical scenes and tightly tied to the stone walls, belong in their large ordering to the line of Correggio and his Baroque followers.

Beyond those classical attributes, the chemical substance salt, was associated with earth and its alchemical symbol was a downward-pointing triangle, bisected by a horizontal line.

If light consisted strictly of ordinary or classical particles, and these particles were fired in a straight line through a slit and allowed to strike a screen on the other side, we would expect to see a pattern corresponding to the size and shape of the slit.

In 1989, von Trier's mother revealed on her deathbed that the man who he thought was his father was not, and that she had had a tryst with her former employer, Fritz Michael Hartmann ( 1909 – 2000 ), who descended from a long line of Roman Catholic classical musicians ( his grandfather was Emil Hartmann, his great grandfather J. P. E.

Typically a 10 pulses per second dial is used into a classic Strowger Exchange and a 20 pps dial would be used into a Crossbar or Electronic exchange where line lengths were reasonable and the receiving register was not limited by classical telephone relays, even with the isthmus armature.

This classical Kelvin – Stokes theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂ Σ:

Their approach was similar to classical music ; each instrument played its own melodic line to generate a grand musical theme.

Though phrases and even sentences from many classical authors are inwoven here and there, the narrative flows easily, with no trace of the jolts and jerks which offend us in almost every line of an imitator of the classics like Sidonius.

The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass.

A 56 kbit / s line is a digital connection capable of carrying 56 kilobits per second ( kbit / s ), or 56, 000 bit / s, the data rate of a classical single channel digital telephone line in North America.

The Aeneid, like other classical epics, is written in dactylic hexameter: each line consists of six metrical feet made up of dactyls ( one long syllable followed by two short syllables ) and spondees ( two long syllables ).

There are devices on the market that combine the functions of an ISDN TA with those of a classical modem ( with an ISDN line interface ).

Bohr never specified a demarcation line above which objects cease to be quantum and become classical.

A classical example of a locally regular space that is not regular is the bug-eyed line.

Instead of taking on the classical criticism of art that involved knowledge of color, composition and line, Gautier was strongly influenced by Denis Diderot's idea that the critic should have the ability to describe the art so as the reader can " see " the art through his description.

In addition to imitations of works by the classical writers Seneca and Horace, he experimented in stanza forms including the rondeau, epigrams, terza rima, ottava rima songs, satires and also with monorime, triplets with refrains, quatrains with different length of line and rhyme schemes, quatrains with codas, and the French forms of douzaine and treizaine.

Another precedent to the prepared piano was an experiment by the French composer Maurice Delage ( 1879 – 1961 ): his Ragamalika ( 1912 – 22 ), based on the classical music of India, calls for a piece of cardboard to be placed under the B-flat in the second line of the bass clef to dampen the sound, imitating the sound of an Indian drum.

Indian classical music ( marga ) is monophonic, and based on a single melody line or raga rhythmically organized through talas.

Bosworth, in line with the epigraphic tradition of modern classical studies, points out that Arrian is a secondary source of Alexander's biographical data: " Arrian is prone to misread and misinterpret his primary sources, and the smooth flow of his narrative can obscure treacherous quicksands of error ".

The next church he designed was St Andrew's Hove, East Sussex, in Waterloo Street, Brunswick, ( 1827 – 28 ); the plan of the building is in line with Georgian architecture, though stylistically the Italianate style was used, the only classical church Barry designed that was actually built.

Romanticism style dominated Norwegian music " until well into the 20th century, whether expressed through modifications to the national Romantic idiom of Grieg or through a more classical / international line " like Catharinus Elling or Halfdan Cleve.