Geometrical Researches on the Theory of ParallelsUniversity of Texas, 1891 - 50 σελίδες |
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Σελίδα 15
... second time ( Theorem 2 ) , therefore it must meet CD somewhere at H ( Theorem 3 ) . Now let E ' be a point on the production of AB and E'K ' perpendic- ular to the production of the line CD ; draw the line E'F ' making so small an ...
... second time ( Theorem 2 ) , therefore it must meet CD somewhere at H ( Theorem 3 ) . Now let E ' be a point on the production of AB and E'K ' perpendic- ular to the production of the line CD ; draw the line E'F ' making so small an ...
Σελίδα 19
... second assumption can likewise be admitted without leading to any contradiction in the results , and founds a new geometric science , to which I have given the name Imaginary Geometry , and which I in- tend here to expound as far as the ...
... second assumption can likewise be admitted without leading to any contradiction in the results , and founds a new geometric science , to which I have given the name Imaginary Geometry , and which I in- tend here to expound as far as the ...
Σελίδα 23
... second time , consequently it must meet EF some- where in H ; therefore AB and EF are mutually parallel . E A Ꮐ FIG . 13 . H D Now let the parallels AB and CD ( Fig . 13 ) lie in two planes whose intersection line is EF . From a random ...
... second time , consequently it must meet EF some- where in H ; therefore AB and EF are mutually parallel . E A Ꮐ FIG . 13 . H D Now let the parallels AB and CD ( Fig . 13 ) lie in two planes whose intersection line is EF . From a random ...
Σελίδα 42
... second , however , we must write correspondingly but we have cos II ( x — c ) = cos ( π — В ) cos ПI ( a ) ; cos II ( x — c ) = — cos II ( c — x ) ( Theorem 23 ) , and also cos ( 7 — B ) = — cos B. If A is a right or an obtuse angle ...
... second , however , we must write correspondingly but we have cos II ( x — c ) = cos ( π — В ) cos ПI ( a ) ; cos II ( x — c ) = — cos II ( c — x ) ( Theorem 23 ) , and also cos ( 7 — B ) = — cos B. If A is a right or an obtuse angle ...
Σελίδα 45
... second . In addition , it is worthy of notice that the four equations ( 8 ) of plane geometry pass over into the equations for spherical triangles , if we put a 1 , b √− 1 , c √- 1 , instead of the sides a , b , c ; with this ...
... second . In addition , it is worthy of notice that the four equations ( 8 ) of plane geometry pass over into the equations for spherical triangles , if we put a 1 , b √− 1 , c √- 1 , instead of the sides a , b , c ; with this ...
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).