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algebra altitude ancient angles are equal applications Archimedes Aristotle attempt axioms base beginning better bisect Book called century A.D. CHAPTER circle circumference congruent considered construction corollary course cube curve cylinder define definition distance drawing easily edge educational elementary geometry equilateral etry Euclid Euclid's Elements example exercises fact figure geom given line Greek Heron of Alexandria hexagon illustration incommensurable inscribed interest intersect isosceles known locus logic mathematician mathematics means measure method modern octahedron parallel parallelepiped parallelogram perpendicular plane geometry Plato polyhedrons postulate practical prism problem Proclus proof proportion proved pupil pyramid Pythagoras Pythagorean Theorem question radius ratio reason regular polygons relating right angles right triangle segment sides solid geometry sphere spherical square statement straight line surface syllabus tangent teacher teaching textbook Thales thing tion to-day trigonometry usually vertex vertices volume words writers
Page 258 - Two triangles having an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles.
Page 248 - If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse : I.
Page 229 - Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
Page 190 - If two triangles have two sides of the one equal, respectively, to two sides of the other, but the included angle of the first triangle greater than the included angle of the second, then the third side of the first is greater than the third side of the second.
Page 144 - But when a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle, and the straight line which stands on the other is called a perpendicular to it (Def.
Page 127 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Page 295 - If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines.
Page 299 - If two angles not in the same plane have their sides respectively parallel and lying on the same side of the straight line joining their vertices, they are equal, and their planes are parallel. Let the corresponding sides of angles A and A' in the planes MN and PQ be parallel, and lie on the same side of AA'.