...three angles in triangle ABC is not greater than two right angles. PROPOSITION XX. THEOREM. In any triangle, the sum of the three angles is equal to two right angles. Having already proved that the sum of the three angles of a triangle cannot exceed two right angles,...

...sum of the angles of the triangle A ADC must be zR + a, which is absurd (by I. above). IV. If in a triangle the sum of the three angles is equal to two right angles, a quadrilateral can always be constructed with four right angles and four equal sides B^ ^ ^O exceeding...

...principle is called a demonstration. Thus in books on Geometry the truth is enunciated that " in any triangle the sum of the three angles is equal to two right angles," and the reasoning is then given at length to prove that this is true. As a knowledge of mechanical...

...parallels, so-called, one on each side. One logical consequence of this is that "if in any rectilineal triangle the sum of the three angles is equal to two right angles, this is also the case for every triangle" — one instance is the criterion for all. As Poincare, perhaps...

...§35, which concludes, "Hence spherical trigonometry is not dependent upon whether in a rectilinear triangle the sum of the three angles is equal to two right angles or not." Just so concludes Chapter XI of his New Elements: "Therefore the equations for spherical triangles...