Geometrical Researches on the Theory of ParallelsUniversity of Texas, 1891 - 50 Seiten |
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Seite 12
... sphere with a plane is a circle . 13. A straight line at right angles to the intersection of two per- pendicular planes , and in one , is perpendicular to the other . 14. In a spherical triangle equal sides lie opposite equal angles ...
... sphere with a plane is a circle . 13. A straight line at right angles to the intersection of two per- pendicular planes , and in one , is perpendicular to the other . 14. In a spherical triangle equal sides lie opposite equal angles ...
Seite 19
... spherical triangle . = d . 23. For every given angle a we can find a line p such that II ( p ) Let AB and AC ( Fig . 10 ) be two straight lines which at the inter . section point A make the acute angle ; take at random on AB a point B ...
... spherical triangle . = d . 23. For every given angle a we can find a line p such that II ( p ) Let AB and AC ( Fig . 10 ) be two straight lines which at the inter . section point A make the acute angle ; take at random on AB a point B ...
Seite 24
... sphere with planes ; in such triangles , therefore , the sides and angles are in contrary order . In the opposite ... sphere a perpendicular dropped whose pro- longations both ways cut both opposite triangles in the points D and D ' of ...
... sphere with planes ; in such triangles , therefore , the sides and angles are in contrary order . In the opposite ... sphere a perpendicular dropped whose pro- longations both ways cut both opposite triangles in the points D and D ' of ...
Seite 25
... spherical triangle Y equals that of the opposite triangle ABC ' , having a side AB in common with the triangle P ... sphere ( Theorem 26 ) . Hence it follows that P + Y = C , and since P + X + Y + Z - π , therefore we have also P ( A ...
... spherical triangle Y equals that of the opposite triangle ABC ' , having a side AB in common with the triangle P ... sphere ( Theorem 26 ) . Hence it follows that P + Y = C , and since P + X + Y + Z - π , therefore we have also P ( A ...
Seite 26
... spherical triangle equals the sum of the two equal angles in the quadrilateral which are not the right angles . Therefore we can , for every spherical triangle , in which the sum of the three angles is S , find a quadrilateral with ...
... spherical triangle equals the sum of the two equal angles in the quadrilateral which are not the right angles . Therefore we can , for every spherical triangle , in which the sum of the three angles is S , find a quadrilateral with ...
Häufige Begriffe und Wortgruppen
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
Beliebte Passagen
Seite 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,...
Seite 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.
Seite 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.
Seite 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.
Seite 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...
Seite 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
Seite 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)...
Seite 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).