EUCLID'S PARALLEL POSTULATE: ITS NATURE, VALIDITY, AND PLACE IN GEOMETRICAL SYSTEMS

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Side 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...
Side 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...
Side 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...
Side 6 - The state of the exact sciences proves, says Mr. Gladstone, that, as respects religion " the association of these two ideas...
Side 31 - ... 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 64 - 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.
Side 107 - Peano defines the straight line ab as a class of points #, such that any point y, whose distances from a and b are respectively equal to the distances of x from a and b, must be coincident with x.
Side 101 - Mill replies, that, without so following them, " we may know that, if they ever do meet, or indeed if, after diverging from one another, they begin again to approach, this must take place, not at an infinite,. but at a finite distance. Supposing, therefore, such to be the case, we can transport ourselves thither in imagination, and can frame a mental image of the appearance which one or both of the lines must present at that point, which we may rely upon as being precisely similar to the reality.
Side 24 - ... axiom. Lobatchewsky's views on the foundation of geometry were first made public in a discourse before the physical and mathematical faculty of the University of Kasan (of which he was then rector), and first printed in the Kasan Messenger for 1829, and then in the Gelehrte Schriften der Universitat Kasan, 1836-1838, under the title "New Elements of Geometry, with a complete theory of parallels.
Side 9 - Euclid's own definition was, that parallel lines are straight lines which lie in the same plane and will not meet however far produced.

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