Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical Systems ...Open Court Publishing Company, 1905 - 192 sider |
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Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical ... John William Withers Uten tilgangsbegrensning - 1905 |
Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical ... John William Withers Uten tilgangsbegrensning - 1905 |
Vanlige uttrykk og setninger
absolute abstract actual experience analytical anguli assumed assumption Bolyai certainly consider construction contradiction Crelle's Journal curved defined definition determined dimensional direction distance elliptic Elliptic geometry empirical ence equal essential etry Euclid Euclidean geometry Euclidean plane Euclidean space Euclidean system exist facts figures Foundations of Geometry fourth dimension free mobility G. B. Halsted Gauss geodesics geom Grundlagen der Geometrie Helmholtz Hence homogeneity idea infinity intersect intuition involved judgment Kant Kant's Klein Leipzig Lobatchewsky logical manifold Math Mathematics meaning metrical geometry nature of space necessary necessity non-Euclid non-Euclidean geometry non-Euclidean systems notion parallel lines parallel postulate Peano peculiar perception philosophical position possible priori problem Proclus projective geometry propositions prove question reality regarded relations requires Riemann right angles Russell Saccheri sense sensory Sophus Lie spatial straight line surface synthetic theorems theory of parallels three dimensions tion transitive relations triangle triangle's angle sum true two-dimensional validity
Populære avsnitt
Side 1 - Assuming as an axiom that two straight lines which cut one another cannot both be parallel to the same straight line ; deduce Euclid's twelfth axiom as a corollary of Euc.
Side 3 - ... case of parallelism, the sum of the interior angles on one side of the transversal must be the same as that upon the other side. 23 Engel and Staeckel op. cit. p. 19. 24
Side 83 - The same truth doubtless suggested itself over and over again to the workmen in clay and stone of Babylonia, Egypt, and Greece in the mosaics and pavements which they are known to have made from differently colored stones of the same shape. In this way, too, it was easily found that the plane " space " about a point can be completely filled only by three kinds of regular polygons, that is, by six equilateral triangles, by four squares and by three regular hexagons," 15 and hence that this space is...
Side 47 - ... mathematicians, notably by Legendre, to give a proof of this proposition ; that is, to show that it is a necessary consequence of the simpler axioms preceding it. Legendre proved that the sum of the angles of a triangle can never exceed two right angles, and that if there is a single triangle in which this sum is equal to two right angles, the same is true of all triangles. This was, of course, on the supposition that a line is of infinite length.
Side 32 - If (n — 1) points of a body remain fixed, so that every other point can only describe a certain curve, then that curve is closed. These axioms, says Helmholtz, suffice to give, with the axiom of three dimensions, the Euclidean and non-Euclidean systems as the only alternatives. That they suffice, mathematically, cannot be denied, but they seem, in some respects, to go too far. In the first place, there...
Side 33 - ... ueberall constant." See also the second hypothesis of Helmholtz. Third, any point may be moved into any other; the free mobility of rigid bodies. If any point remains at rest any region in which it is may be moved about it in innumerable ways, and so that any point other than the one at rest may recur. If two points are fixed motion is still possible in a specific way. Three fixed points not costraight prevent all motion (p. 446, § 5). Thus we have the third assumption of Helmholtz, combined...
Side 21 - ... particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely...
Side 54 - The geometer of to-day knows nothing about the nature of actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or a future eternity.