...acute-angled 2 (A) Oftliefir»! Properties, and of Trianglet winch are equal in all respects. (a) Anyone of the angles of a triangle is less than two right angles cor. 5 (A) Triangles which have two sides and the included angle of the one equal to two sides, and...

...acute-angled 2 (A) Ot'lltrßrsl Properties, and of Triangles vhu-h are equal in all respeits. (aj Any one of the angles of a triangle is less than two right angles cor. 5 (б) Triangles which have two sides and the included angle of the one equal to two sides, and...

...angle between these two depends in a certain way upon the distance of the point from the line. The sum of the angles of a triangle is less than two right angles by a quantity propoilional to the area of the triangle. The whole of this geometry is worked out in...

...say, that more than one parallel can be drawn to a given line through a given point, or that the sum of the angles of a triangle is less than two right angles. The question of the character of the socalled geometrical axioms thus assumes an aspect very different...

...as far as they cover the same ground, are identical in substance, though different in form. The sum of the angles of a triangle is less than two right angles, so that a rectangle is impossible ; the angle-sums of two triangles of equal area are equal ; no two...

...angles we can show that then only one parallel is possible. It follows that in our new geometry the sum of the angles of a triangle is less than two right angles. It is easy to prove also that if two lines which meet a third so as to make the sum of the interior...

...say, that more than one parallel can be drawn to a given line through a given point, or that the sum of the angles of a triangle is less than two right angles. The question of the character of the socalled geometrical axioms thus assumes an aspect very different...

...pseudo-spherical geometry — (a.) Theorems which assume that non-parallel lines must meet are not true. (*.) Theorems involving parallelism must conform to the...angles — that is, by its pseudo-spherical deficiency. PSEUDO-SPHERE 112. Remark. — The circumference of a circle on the plane surface =L2irr; on the spherical...

...the plane, spherical, and pseudo-spherical surfaces respectively is shown in the above pictures. 111. Pseudo-spherical geometry can be built up from the...angles — that is, by its pseudo-spherical deficiency. CYLINDHR PSEUDO'SPHERE 112. Remark. — The circumference of a circle on the plane surface = 27rr;...

...angle between these two depends in a certain way upon the distance of the point from the line. The sum of the angles of a triangle is less than two right angles by a quantity proportional to the area of the triangle. The whole of this geometry is worked out in...