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.
  • 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.
  • 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.
  • 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.