Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical Systems ...Open Court Publishing Company, 1905 - 192 σελίδες |
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Σελίδα 10
... hold for Euclidean geometry ; they are not true for pseudo - spherical space where parallel lines are still possible and where Euclid's definition is still valid . The objection to the third definition is its use of the term " direction ...
... hold for Euclidean geometry ; they are not true for pseudo - spherical space where parallel lines are still possible and where Euclid's definition is still valid . The objection to the third definition is its use of the term " direction ...
Σελίδα 13
... hold in elliptic space and also involves the untenable assertion that , in the 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 ...
... hold in elliptic space and also involves the untenable assertion that , in the 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 ...
Σελίδα 29
... hold that this line would be straight ; on the contrary it would be an hyperbola as in the perspective of the terrestrial horizon . If we accept Riemann's hypothesis we cannot be sure that there will be any such line at all , for we do ...
... hold that this line would be straight ; on the contrary it would be an hyperbola as in the perspective of the terrestrial horizon . If we accept Riemann's hypothesis we cannot be sure that there will be any such line at all , for we do ...
Σελίδα 39
... holds for the surface of the sphere . If the measure of cur- vature has in actual space a positive value howso- ever small there are in reality no such things as parallel lines , for all lines meet if sufficiently pro- duced , and space ...
... holds for the surface of the sphere . If the measure of cur- vature has in actual space a positive value howso- ever small there are in reality no such things as parallel lines , for all lines meet if sufficiently pro- duced , and space ...
Σελίδα 45
... hold good for corresponding theorems in the other two it afforded , for the first time , a conclusive proof that the geometries of Lobatchewsky and Riemann are no more contradictory than is Euclid itself , and thus gave to all three ...
... hold good for corresponding theorems in the other two it afforded , for the first time , a conclusive proof that the geometries of Lobatchewsky and Riemann are no more contradictory than is Euclid itself , and thus gave to all three ...
Άλλες εκδόσεις - Προβολή όλων
Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical ... John William Withers Δεν υπάρχει διαθέσιμη προεπισκόπηση - 2016 |
Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical ... John William Withers Δεν υπάρχει διαθέσιμη προεπισκόπηση - 2018 |
Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical ... John Withers Δεν υπάρχει διαθέσιμη προεπισκόπηση - 2017 |
Συχνά εμφανιζόμενοι όροι και φράσεις
absolute abstract ALFRED BINET analytical anguli apodeictic assumed assumption body Bolyai certainly Cloth conception consider construction Crelle's Journal curved defined definition determined direction distance Edition Elliptic geometry empirical ence equal essential etry Euclid Euclidean geometry Euclidean plane Euclidean space Euclidean system facts figures Foundations of Geometry free mobility G. B. Halsted Gauss geodesics geom Grundlagen der Geometrie Helmholtz Hence homogeneity idea intuition involved judgment Kant Kant's Klein Leipzig Lobatchewsky logical manifold Math Mathematics meaning measure of curvature nature of space necessary non-Euclid Non-Euclidean Geometry non-Euclidean systems notion Pages parallel lines parallel postulate PAUL CARUS Peano peculiar perception philosophical plane possible principle priori problem Proclus projective geometry propositions prove Psychology question reality regarded relations Riemann right angles Russell Saccheri Science sense sensory Sophus Lie spatial straight line surface synthetic theorems theory of parallels three dimensions tion triangle triangle's angle sum true two-dimensional validity
Δημοφιλή αποσπάσματα
Σελίδα 4 - If a straight line meets two straight lines, so as to " make the two interior angles on the same side of it taken " together less than two right angles...
Σελίδα 11 - 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.
Σελίδα 25 - AF, to the not-cutting lines, as AG, we must come upon a line AH, parallel to DC, a boundary line, upon one side of which all lines AG are such as do not meet the line DC, while upon the other side every straight line AF cuts the line DC. The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism), which we will here designate by f] (p) for AD = p. If H (p) is a right angle, so will the prolongation AE...
Σελίδα 13 - CD are crossed by a transversal (see figure on page 42), if the sum of the interior angles on one side of the transversal is...
Σελίδα 44 - Whitehead in his Universal Algebra, § 262, recurs to this important point, saying, "The idea of a space of one type as a locus in space of another type, and of dimensions higher by one, is due partly to J. Bolyai, and partly to Beltrami. Bolyai points out that the relations between lines formed by great circles on a two-dimensional limit-surface are the same as those of straight lines in a Euclidean plane of two dimensions. "Beltrami proves, by the use of the pseudosphere, that a...
Σελίδα 6 - The state of the exact sciences proves, says Mr. Gladstone, that, as respects religion " the association of these two ideas...