Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical Systems ...Open Court Publishing Company, 1905 - 192 σελίδες |
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Σελίδα 2
... determination by Menæchmus of the necessary conditions for the invertibility of a theorem which affords a fruitful method of enlarging the number of propositions , the introduction of formal logic and the powerful influence of the ...
... determination by Menæchmus of the necessary conditions for the invertibility of a theorem which affords a fruitful method of enlarging the number of propositions , the introduction of formal logic and the powerful influence of the ...
Σελίδα 31
... determined . These matters of fact are , like all mat- ters of fact , not necessary but only of empirical certainty . " Riemann introduces into the problem the gen- eral conception of a manifold of which space is but a specialization ...
... determined . These matters of fact are , like all mat- ters of fact , not necessary but only of empirical certainty . " Riemann introduces into the problem the gen- eral conception of a manifold of which space is but a specialization ...
Σελίδα 33
... determination of the line comes in part to be a matter of giving these quantities as functions of one variable . The problem then is to establish a mathematical expression for the length of a line and to this end the quantities x must ...
... determination of the line comes in part to be a matter of giving these quantities as functions of one variable . The problem then is to establish a mathematical expression for the length of a line and to this end the quantities x must ...
Σελίδα 34
... determining the circle which most nearly coincides with the curve at the point consid- ered . This circle will pass through three consecu- If tive points of the curve and hence its construction 34 DISCOVERY OF NON - EUCLIDEAN SYSTEMS.
... determining the circle which most nearly coincides with the curve at the point consid- ered . This circle will pass through three consecu- If tive points of the curve and hence its construction 34 DISCOVERY OF NON - EUCLIDEAN SYSTEMS.
Σελίδα 35
... determined the curvature of surfaces by their amount of departure from the plane . In the case of curved surfaces we can draw through any point an unlimited number of geodesic lines whose curva- ture , generally speaking , will not be ...
... determined the curvature of surfaces by their amount of departure from the plane . In the case of curved surfaces we can draw through any point an unlimited number of geodesic lines whose curva- ture , generally speaking , will not be ...
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...