Geometrical Researches on the Theory of ParallelsOpen Court Publishing Company, 1914 - 50 Seiten |
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acute angle angle BAC angles equal angles opposite assume axes AA axioms axis BF BC the perpendicular Bolyai boundary line boundary-surface CD somewhere circle congruent consequently cos II curvature cut the line dicular draw end-points equal angles equations erected Euclid Euclidean geometry Foundations of Geometry Gauss given Greek Hence hypothenuse II(a II(b II(c imaginary geometry intersection point Kasan let fall line AB line CD line DC Lobachevski mid-point Non-Euclidean Geometry opposite angles opposite triangles parallel to DD parallels AA pendicular perpen perpendicu perpendicular HK perpendicular sides pertain plane geometry point F prolongation quadrilateral random rectilineal triangle right angles right-angled triangles sin II small soever soever the angle space sphere spherical triangle spherical trigonometry straight line Theo Theorem 16 Theorem 23 theory of parallels three angles triangle ABC Fig triangle equal triangle the sum ular whence follows
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Seite 14 - K'AE, H'AE' to the non-intersecting. In accordance with this, for the assumption /7(p) = Mтr the lines can be only intersecting or parallel; but if we assume that /7(p) < 'Лтг, then we must allow two parallels, one on the one and one on the other side; in addition we must distinguish the remaining lines into non-intersecting and intersecting. For both assumptions it serves as the mark of parallelism that the line becomes intersecting for the smallest deviation toward the side where lies the parallel,...
Seite 6 - 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...
Seite 14 - AG, if they lie upon the other sides of the parallels AH and AK, in the opening of the two angles EAH = $7r— H(p), E'AK = Jir — H(/»). between the parallels and EE' the perpendicular to AD. Upon the other side of the perpendicular EE' will in like manner the prolongations AH
Seite 13 - A meet the line DC, as for example AF, or some of them, like the perpendicular AE, will not meet the line DC. In the uncertainty whether the perpendicular AE is the only line which does not meet DC, we will assume it may be possible that there are still other lines, for example AG, which do not cut DC, how far soever they may be prolonged. In passing over from the cutting lines, as 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...
Seite 13 - DAK = 77 (p) will lie also a line AK, parallel to the prolongation DB of the line DC, so that under this assumption we must also make a distinction of sides in parallelism.
Seite 11 - VSKY 139 1. A straight line fits upon itself in all its positions. By this I mean that during the revolution of the surface containing it the straight line does not change its place if it goes through two unmoving points in the surface: (ie, if we turn the surface containing it about two points of the line, the line does not move.) 2.
Seite 3 - It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration, and that Lobatschewsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or, say, a system of non-Euclidian plane geometry.
Seite 18 - ... 21. From a given point we can always draw a straight line that shall make with a given straight line an angle as small as we choose.
Seite 11 - In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it has come to us from Euclid.
Seite 48 - At once the supposed fact that our space does not interfere to squeeze us or stretch us when we move, is envisaged as a peculiar property of our space. But is it not absurd to speak of space as interfering with anything? If you think so, take a knife and a raw potato, and try to cut it into a seven-edged solid.