Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical Systems ...Open Court Publishing Company, 1905 - 192 σελίδες |
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Σελίδα 35
... curved surfaces we can draw through any point an unlimited number of geodesic lines whose curva- ture , generally speaking , will not be the same . then we draw all the geodesics from the point at which the curvature is tested to the ...
... curved surfaces we can draw through any point an unlimited number of geodesic lines whose curva- ture , generally speaking , will not be the same . then we draw all the geodesics from the point at which the curvature is tested to the ...
Σελίδα 48
... curved line between any two points could be measured with this rule , and always with the same determinate result , no matter from what point in the line we begin . Of course this distance will not be what we usually mean by the term ...
... curved line between any two points could be measured with this rule , and always with the same determinate result , no matter from what point in the line we begin . Of course this distance will not be what we usually mean by the term ...
Σελίδα 117
... say , for 37 We can not say the opposite end for the lines in each case and in both directions are supposed to be unbounded . in that event they would necessarily be curved or at PSYCHOLOGIC ASPECTS OF THE PROBLEM 117.
... say , for 37 We can not say the opposite end for the lines in each case and in both directions are supposed to be unbounded . in that event they would necessarily be curved or at PSYCHOLOGIC ASPECTS OF THE PROBLEM 117.
Σελίδα 118
... curved or at any rate not continuous straight lines . Even the assumption which leads to Euclid seems not to be in accord with the lines of experience which we call parallel ; for all such lines must turn a little way after ceasing to ...
... curved or at any rate not continuous straight lines . Even the assumption which leads to Euclid seems not to be in accord with the lines of experience which we call parallel ; for all such lines must turn a little way after ceasing to ...
Σελίδα 119
... curved , are not at all familiar to us . But suppose these saddled shaped surfaces had been the ones with which our notion of curved lines had always been associated and that pseudo - spheres and spheres had simply exchanged places in ...
... curved , are not at all familiar to us . But suppose these saddled shaped surfaces had been the ones with which our notion of curved lines had always been associated and that pseudo - spheres and spheres had simply exchanged places in ...
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...