(1) The falseness of the idea of principle, is typified by a Cartesian or Euclidean geometry.

(2) Mapping it onto the Earth's surface is far more complex, however, because there may be little relationship between proximity in Euclidean geographic space and positionality.

(3) Then like in the linear separable case, it finds the optimal separating hyper-plane in the Hilbert space H that would correspond to a nonlinear boundary in the original Euclidean space.

(4) From this point of view, Euclidean geometry is a very favorable place to begin a student's serious mathematical training.

(5) This is, of course, how Beltrami first showed that hyperbolic geometry was no less consistent than Euclidean geometry (though he used a different model).

(6) It seemed to me that I could do some useful work in giving the student a historical perspective and in showing how the multitude of abstract concepts have arisen and are present in Euclidean spaces.

(7) It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry.

(8) Mathematics has considered alternatives to Euclidean space since the early nineteenth century.

(9) By that I mean that they would be chunks of familiar Euclidean space; one could require them to be cuboids, but this is not very important mathematically.

(10) He was one of the earliest mathematicians to demonstrate that the ordinary experience of Euclidean concepts can be extended meaningfully beyond geometry into the idealised constructions of more complex abstract mathematics.

(11) Obviously, when no obstacles are used, then the matrix represents a Euclidean space with dimensionality equal to two.

(12) Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.

(13) Today we call these three geometries Euclidean , hyperbolic, and absolute.

(14) It is important to note, however, that the extra dimension, though curled up, was still Euclidean in nature.

(15) The approach that concentrates on non-Euclidean geometry is ideal for students who already have a mastery of Euclidean geometry, but it cannot replace such a mastery.

(16) Multidimensional scaling analyses were used to represent the relationships of the data set in n-dimensional Euclidean space in an attempt to identify putative group structures.

(17) Well, look at Cartesian geometry: In a Cartesian geometry - or Euclidean , which are interchangeable, in one sense - you have certain assumptions.

(18) To sum up, I am asserting that Euclidean geometry is the only mathematical subject that is really in a position to provide the grounds for its own axiomatic procedures.

(19) The latter are mediated by DNA-loops bringing two chemically remote segments of the DNA close in Euclidean space.

(20) This is a peer reviewed journal devoted to the Euclidean geometry.