...angles, and that a geometry may exist, if not in nature at least in analysis, on the hypothesis that the sum of the angles is less than two right angles ; and he accordingly attempts to establish such a geometry, viz. a, b, c being the sides of a rectilinear...

...angles, and that a geometry may exist, if not in nature at least in analysis, on the hypothesis that the sum of the angles is less than two right angles ; and he accordingly attempts to establish such a geometry, viz. a, b, с being the sides of a rectilinear...

...noted that these formulas state the relations existing among th sides and the angles of a triangle in which the sum of the angles is less than two right angles. Lobatschefskij, Professor of Mathematics in the University of Kasan, and Johann Bolyai, Lieutenant...

...transversal together less than two right angles, the lines if sufficiently produced will meet on the side on which the sum of the angles is less than two right angles. With these three theorems proved, which is easily accomplished, particularly if Playfair's axiom is...

...equal to tim right angles, then in every triangle the sum of the angles must likvinse be equal to Iwo right angles. This proposition was already proved...triangles, in each of which the sum of the angles = IT. L»gendre makes an attempt to prove that the ?um of the angles of a triangle is equal to two...

...in which the sum of the angles is equal to two right angles, but we also know curvilinear triangles in which the sum of the angles is less than two right angles ; if we call the sides of the first straight lines, we are adopting Euclidean geometry ; if we apply...

...C3', cutting C' orthogonally. Hence its centre lies outside C'. We thus obtain a curvilinear triangle in which the sum of the angles is less than two right angles ; and since the angles in this triangle are equal to those in the nominal triangle, our result is proved....

...side is less than two right angles, then the lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles. If you draw a picture, the statement, in spite of its length, becomes quite clear. It implies, in particular,...

...less than two right angles, then the straight lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles. In other words, referring to the figure, if < A + < B < 180° the lines L, and L2 will intersect at...

...C,', cutting C' at right angles. Hence its centre lies outside C'. We thus obtain a " triangle " of which the sum of the angles is less than two right angles, and since these angles are equal to the angles of the ideal triangle, this result also holds for the ideal...