## Euclidean Sentence Examples

- A much less wise class than the 7r-computers of modern times are the pseudo-circle-squarers, or circle-squarers technically so called, that is to say, persons who, having obtained by illegitimate means a
**Euclidean**construction for the quadrature or a finitely expressible value for 7r, insist on using faulty reasoning and defective mathematics to establish their assertions. - Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-
**Euclidean**geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces. - For the subjects under this heading see the articles CONIC SECTIONS; CIRCLE; CURVE; GEOMETRICAL CONTINUITY; GEOMETRY, Axioms of; GEOMETRY,
**Euclidean**; GEOMETRY, Projective; GEOMETRY, Analytical; GEOMETRY, Line; KNOTS, MATHEMATICAL THEORY OF; MENSURATION; MODELS; PROJECTION; Surface; Trigonometry. - At the same time, it delights the pure theorist by the simplicity of the logic with which the fundamental theorems may be established, and by the elegance of its mathematical operations, insomuch that hydrostatics may be considered as the
**Euclidean**pure geometry of mechanical science. - Reference should be made to the article Geometry:
**Euclidean**, for a detailed summary of the**Euclidean**treatment, and the elementary properties of the circle. - The problem of finding a square equal in area to a given circle, like all problems, may be increased in difficulty by the imposition of restrictions; consequently under the designation there may be embraced quite a variety of geometrical problems. It has to be noted, however, that, when the " squaring " of the circle is especially spoken of, it is almost always tacitly assumed that the restrictions are those of the
**Euclidean**geometry. - Since the area of a circle equals that of the rectilineal triangle whose base has the same length as the circumference and whose altitude equals the radius (Archimedes, KIKXou A ir, prop.i), it follows that, if a straight line could be drawn equal in length to the circumference, the required square could be found by an ordinary
**Euclidean**construction; also, it is evident that, conversely, if a square equal in area to the circle could be obtained it would be possible to draw a straight line equal to the circumference. - Other words, by a non-
**Euclidean**construction he trisected the angle AOC, for it is readily seen that, since FD' = FO = OC, the angle FOB = 2 AOC. 6 This couplet of constructions is as important from the calculator's point of view as it is interesting geometrically. - Montucla, successful attempt to show that quadrature of the circle by a
**Euclidean**construction was impossible.' - In 1873 Charles Hermite proved that the base of the Napierian logarithms cannot be a root of a rational algebraical equation of any degree.3 To prove the same proposition regarding 7r is to prove that a
**Euclidean**construction for circle-quadrature is impossible. - A discussion of these concepts and the various definitions of angles in
**Euclidean**geometry is to be found in W. - A special application of his theory of continuous groups was to the general problem of non-
**Euclidean**geometry. - Spinoza's philosophy is expounded ordine geometrico and with
**Euclidean**cogency from a relatively small number of definitions, axioms and postulates. - He was much interested, too, in universal algebra, non-
**Euclidean**geometry and elliptic functions, his papers "Preliminary Sketch of Bi-quaternions" (1873) and "On the Canonical Form and Dissection of a Riemann's Surface" (1877) ranking as classics. - (1) Generation of the concept through imaginaries and development into a method applicable to
**Euclidean**geometry. - 0) 2 = i suitable for non-
**Euclidean**space, and w 2 = o suitable for**Euclidean**space; we confine ourselves to the second, and will call the indicated bi-quaternion p+wq an octonion. - To fix a weighted point and a weighted plane in
**Euclidean**space we require 8 scalars, and not the 12 scalars of a tri-quaternion.