At the turn of the 20th century, Otto Stolz, Paul du Bois-Reymond, Giuseppe Veronese, and others produced controversial work on non-Archimedean models of Euclidean geometry, in which the distance between two points may be infinite or infinitesimal, in the Newton – Leibniz sense.

Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain sense: there is an algorithm that, for every proposition, can be shown either true or false.

Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.

General vectors in this sense are fixed-size, ordered collections of items as in the case of Euclidean vectors, but the individual items may not be real numbers, and the normal Euclidean concepts of length, distance and angle may not be applicable.

Dedekind used the German word Schnitt ( cut ) in a visual sense rooted in Euclidean geometry.

Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point ( a tangent vector ).

The equivalents of lines are not defined in the usual sense of " straight line " in Euclidean geometry but in the sense of " the shortest paths between points ," which are called geodesics.

In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry.

In these coordinates, the metric tensor is well-approximated by the Euclidean metric, in the precise sense that

The n-sphere is a locally conformally flat manifold that is not globally conformally flat in this sense, whereas a Euclidean space, a torus, or any conformal manifold that is covered by an open subset of Euclidean space is ( globally ) conformally flat in this sense.

In this sense, it could be argued that classical painting, with its immobile perspective and Euclidean geometry, was an abstraction, not an accurate representation of the real world.

For example, a circle is a concept that makes sense in Euclidean geometry, but not in affine linear geometry or projective geometry, where circles cannot be distinguished from ellipses, since one may squeeze a circle to an ellipse.

Thus, in Euclidean geometry three non-collinear points determine a circle ( as the circumcircle of the triangle they define ), but four points in general do not ( they do so only for cyclic quadrilaterals ), so the notion of " general position with respect to circles ", namely " no four points lie on a circle " makes sense.

Finsler manifolds non-trivially generalize Riemannian manifolds in the sense that they are not necessarily infinitesimally Euclidean.

In classical MDS, this norm is the Euclidean distance, but, in a broader sense, it may be a metric or arbitrary distance function.

The affine subspaces are model surfaces — they are the simplest surfaces in R < sup > 3 </ sup >, and are homogeneous under the Euclidean group of the plane, hence they are Klein geometries in the sense of Felix Klein's Erlangen programme.

In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry.

Here d < sub > H </ sub > denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i. e. it must preserve all distances, not only infinitesimally small ones ; for example no compact Riemannian manifold of negative sectional curvature admits such an embedding into Euclidean space.

A topological space X is called simply connected if it is path-connected and any continuous map f: S < sup > 1 </ sup > → X ( where S < sup > 1 </ sup > denotes the unit circle in Euclidean 2-space ) can be contracted to a point in the following sense: there exists a continuous map F: D < sup > 2 </ sup > → X ( where D < sup > 2 </ sup > denotes the unit disk in Euclidean 2-space ) such that F restricted to S < sup > 1 </ sup > is f.

His designs tend to include a strong sense of geometry, often being based on very simple shapes, yet creating unique volumes of space.

Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.

In some sense they describe the “ square root ” of geometry and, just as understanding the square root of-1 took centuries, the same might be true of spinors.

In this sense the minor arc is analogous to “ straight lines ” in spherical geometry.

There is no longer an assumption that axioms are " true " in any sense ; this allows for parallel mathematical theories built on alternate sets of axioms ( see Axiomatic set theory and Non-Euclidean geometry for examples ).

These factors include the number of turns in the sense winding, magnetic permeability of the core, sensor geometry and the gated flux rate of change with respect to time.

If all axioms defining a class of algebras are identities, then the class of objects is a variety ( not to be confused with algebraic variety in the sense of algebraic geometry ).

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 ).

:... projective geometry is simpler than algebra in a certain sense, because we use only five geometric axioms to derive the nine field axioms.

A groundlaying book in the subject by Preparata and Shamos dates the first use of the term " computational geometry " in this sense by 1975.

* " The axiom of choice must be wrong because it implies the Banach-Tarski paradox, meaning that geometry contradicts common sense.

In the old topological literature " compact " was taken to include the Hausdorff property, and this convention is still honored in algebraic geometry ; therefore compactness in the modern sense is called " quasicompactness " in algebraic geometry.

This sense of moderation and fairness is superbly exemplified in an exchange of letters between John Jay and a Tory refugee, Peter Van Schaack.

The process stipulates that the choreographer sense the quality of the initial movement he has discovered and that he feel the rightness of the quality that is to follow it.

It is not a mess you can make sense of ''.

What is the history of criticism but the history of men attempting to make sense of the manifold elements in art that will not allow themselves to be reduced to a single philosophy or a single aesthetic theory??

In addition, they have been converted to Zen Buddhism, with its glorification of all that is `` natural '' and mysteriously alive, the sense that everything in the world is flowing.

Piepsam is not, certainly, religious in any conventional sense.

Mimesis here is not to be confused with literalism or realism in the conventional sense.

Neither is primary experience understood according to the attitude of modern empiricism in which nothing is thought to be received other than signals of sensory qualities producing their responses in the appropriate sense organs.

These desires presuppose a sense of causally efficacious powers in which one is involved, some working for one's good, others threatening ill.

he is questioning, also, every epistemology which stems from Hume's presupposition that experience is merely sense data in abstraction from causal efficacy, and that causal efficacy is something intellectually imputed to the world, not directly perceived.

As long as perception is seen as composed only of isolated sense data, most of the quality and interconnectedness of existence loses its objectivity, becomes an invention of consciousness, and the result is a philosophical scepticism.

The corporation in America is in reality our form of socialism, vying in a sense with the other socialistic form that has emerged within governmental bureaucracy.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

In a sense, Einstein's theory is simpler than Newton's, and there is a corresponding sense in which Copernicus' theory is simpler than Ptolemy's.

The only way to describe Paula Sandburg is to say she is beautiful in a Grecian sense.