Geometrical Researches on the Theory of ParallelsUniversity of Texas, 1891 - 50 σελίδες |
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Σελίδα 12
... spherical triangle equal sides lie opposite equal angles , and inversely . 15. Spherical triangles are congruent ( or symmetrical ) if they have two sides and the included angle equal , or a side and the adjacent angles equal . From ...
... spherical triangle equal sides lie opposite equal angles , and inversely . 15. Spherical triangles are congruent ( or symmetrical ) if they have two sides and the included angle equal , or a side and the adjacent angles equal . From ...
Σελίδα 19
... triangle ADE the angle AED be either a or less ( Theo- rems 8 and 20 ) . Continuing thus we finally attain to such ... spherical triangle . = d . 23. For every given angle a we can find a line p such that II ( p ) Let AB and AC ( Fig ...
... triangle ADE the angle AED be either a or less ( Theo- rems 8 and 20 ) . Continuing thus we finally attain to such ... spherical triangle . = d . 23. For every given angle a we can find a line p such that II ( p ) Let AB and AC ( Fig ...
Σελίδα 24
... triangle at the points A ' , B ' , C ' . Through the three points A , B , C , suppose a plane passed , and upon it from the center of the sphere a perpendicular dropped whose pro- longations both ways cut both opposite triangles in the ...
... triangle at the points A ' , B ' , C ' . Through the three points A , B , C , suppose a plane passed , and upon it from the center of the sphere a perpendicular dropped whose pro- longations both ways cut both opposite triangles in the ...
Σελίδα 25
... spherical triangle Y equals that of the opposite triangle ABC ' , having a side AB in common with the triangle P , and whose third angle C ' lies at the end - point of the diameter of the sphere which goes from C through the center D of the ...
... spherical triangle Y equals that of the opposite triangle ABC ' , having a side AB in common with the triangle P , and whose third angle C ' lies at the end - point of the diameter of the sphere which goes from C through the center D of the ...
Σελίδα 26
... triangle FAD DBH , and then taking away the triangle CGE Supposing in the spherical quadrilateral AFGC a great circle passed through the points A and G , as also through F and C , then will their arcs between AG and FC equal one another ...
... triangle FAD DBH , and then taking away the triangle CGE Supposing in the spherical quadrilateral AFGC a great circle passed through the points A and G , as also through F and C , then will their arcs between AG and FC equal one another ...
Συχνά εμφανιζόμενοι όροι και φράσεις
Alhambra angles equal arcs assumption aura axes axioms axis Belvidere Bolyai boundary line Calcul de variations chord circle congruent consequently curvature draw end-points équation equations erected Euclid Euclid's first Gauss géométrie geometry given greater Greek HALSTED Hence hongrois II(a II(c intersection Johann Bolyai l'axiôme XI less let fall ligne likewise line DC Lobatschewsky logarithmes logarithmes naturels make Maros Vásárhely meet mid-point order pendicular perpendicular pertain produced quadrilateral quelconque rectiligne rectilineal triangle right-angled triangles same says sides small somewhere somme des angles space sphere spherical triangle straight line surface système take tang Temesvár Tentamen Theorem 16 Theorem 23 Théorème de Taylor theory of parallels third three angles triangle ABC Fig Trigonométrie sphérique two right angles Two straight lines University whence follows
Δημοφιλή αποσπάσματα
Σελίδα 4 - 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,...
Σελίδα 3 - 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.
Σελίδα 3 - 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.
Σελίδα 49 - ... 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.
Σελίδα 2 - 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...
Σελίδα 4 - 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
Σελίδα 3 - 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)...
Σελίδα 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).