Geometrical Researches on the Theory of Parallels

University of Texas, 1891 - 50 sider

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Side 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...
Side 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,...
Side 13 - All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes — into cutting and not-cutting. The boundary lines of the one and the other class of those lines will be called parallel to the given line.
Side 13 - EE', all others, if they are sufficiently produced both ways, must intersect the line BC. If !!(/>) < -JTT, then upon the other side of AD, making the same angle KAD = H(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.
Side 59 - ... opposés, si ce n'est que, sur la sphère, les côtés sont réels, et que dans le plan on doit les considérer comme imaginaires, de même que si le plan était une sphère imaginaire.
Side 12 - This holds of plane rectilineal angles among themselves, as also of plane surface angles: (te, dihedral angles.) 7. Two straight lines can not intersect, if a third cuts them at the same angle. 8. In a rectilineal triangle equal sides lie opposite equal angles, and inversely. 9. In a rectilineal triangle, a greater side lies opposite a greater angle. In a right-angled triangle the hypothenuse is greater than either of the other sides, and the two angles adjacent to it are acute. 10. Rectilineal triangles...
Side 14 - EE' the perpendicular to AD. Upon the other side of the perpendicular EE' will in like manner the prolongations AH' and AK' of the parallels AH and AK likewise be parallel to BC; the remaining lines pertain, if in the angle K'AH', to the intersecting, but if in the angles K'AE, H'AE
Side 13 - FIG. 1. 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 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)...
Side 11 - 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).
Side 8 - Lobatschewsky aucun fait nouveau pour moi; mais l'exposition est toute différente de celle que j'avais projetée, et l'auteur a traité la matière de main de maître et avec le véritable esprit géométrique. Je crois devoir appeler votre attention sur ce livre, dont la lecture ne peut manquer de vous causer le plus vif plaisir.

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