# Euclidean Sentence Examples

- Reference should be made to the article Geometry:
**Euclidean**, for a detailed summary of the**Euclidean**treatment, and the elementary properties of the circle. - 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. - 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. - 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. - 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. - (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. - 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.