Euclid's Parallel Postulate: Its Nature, Validity, and Place in Geometrical Systems ...Open Court Publishing Company, 1905 - 192 σελίδες |
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Σελίδα
... prove the Parallel Postulate on a basis of Euclid's other assumptions . II . THE DISCOVERY AND DEVELOPMENT OF NON- EUCLIDEAN SYSTEMS . The clue . Invention of consistent geome- tries independent of the Parallel Postulate . The problem ...
... prove the Parallel Postulate on a basis of Euclid's other assumptions . II . THE DISCOVERY AND DEVELOPMENT OF NON- EUCLIDEAN SYSTEMS . The clue . Invention of consistent geome- tries independent of the Parallel Postulate . The problem ...
Σελίδα 2
... proved . The selection and arrangement of the 1 Gow's History of Greek Geometry , Cambridge , 1884 , p . 198 . 2 Proclus , at close of the Eudemian Summary . Eudemas , a pupil of Aristotle , wrote a history of Geometry which has been ...
... proved . The selection and arrangement of the 1 Gow's History of Greek Geometry , Cambridge , 1884 , p . 198 . 2 Proclus , at close of the Eudemian Summary . Eudemas , a pupil of Aristotle , wrote a history of Geometry which has been ...
Σελίδα 5
... proved , and actually attempts to prove it in book I , proposition 4. He also gives the parallel " axiom " as postulate 5 , but claims that it , too , should be proved , and quotes Germinus 10 in support of this view . Thus it appears ...
... proved , and actually attempts to prove it in book I , proposition 4. He also gives the parallel " axiom " as postulate 5 , but claims that it , too , should be proved , and quotes Germinus 10 in support of this view . Thus it appears ...
Σελίδα 6
... prove it . Euclid makes no attempt as did later writers to conceal the diffi- culty under the cloak of a subtle phraseology . He states it frankly as a petitio principii of the baldest type . It must then have appeared to him not as an ...
... prove it . Euclid makes no attempt as did later writers to conceal the diffi- culty under the cloak of a subtle phraseology . He states it frankly as a petitio principii of the baldest type . It must then have appeared to him not as an ...
Σελίδα 7
... prove it nor proceed any further without it . Moreover , the astronomical system of Eudoxus and the writings of Antolycus make it also apparent that Euclid must have had some knowledge of sur- face spherics and was therefore familiar ...
... prove it nor proceed any further without it . Moreover , the astronomical system of Eudoxus and the writings of Antolycus make it also apparent that Euclid must have had some knowledge of sur- face spherics and was therefore familiar ...
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...