Non-Euclidean Geometry |
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Common terms and phrases
AC and BD acute angle Analytic Geometry angle of parallelism axes axiom axis Beman and Smith's boundary-curve boundary-surface Byerly's circle common perpendicular Corollary cos² curve cut off equal deficiency or excess dicular diedral diminish indefinitely draw Elliptic Geometry equal angles equidistant equidistant-curve Euclid Euclidean Geometry exterior angle finite formulæ four right angles given line greater H moves hypothenuse intersect less limiting position line joining middle point move off indefinitely Non-Euclidean Geometry obtuse angle origin parallel lines parallel to CD pendicular perpen perpendicular erected perpendicular to CD Plane and Solid plane triangles polar equation pole polygon Proof propositions prove quadrilateral radius vector ratio rectangle rectangular coördinates restricted portion revolved right triangle rotation sin² Solid Geometry sphere Spherical Trigonometry straight line tan II tan² tan² iy Theorem three hypotheses triangle ABC values vertex
Popular passages
Page 7 - If two triangles have two sides of one equal respectively to two sides of the other, but the included angle of the first 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 9 - ... perpendicular, they are unequal, and the more remote is the greater ; and conversely, if two oblique lines drawn from a point in a perpendicular are unequal, the greater cuts off a greater distance from the foot of the perpendicular. 9. Theorem. If a perpendicular is erected at the middle point of a straight line, any point not in the perpendicular is nearer that extremity of the line which is on the same side of the perpendicular. Corollary. Two points equidistant from the extremities of a straight...
Page 60 - That is, cos a' = cos b' cos c' + sin 6' sin c' cos A'. .: cos (180° - A) = cos (180° - B) cos (180° - C) + sin (180° - B) sin (180° - '7' cos (180° - a). [Art. 16. d.] .: — cos A = (— cos B) (— cos C) + sin B sin C(— cos a).
Page 24 - Let x and y be any two acute angles, and draw the figures used to prove the formulae for the sine and cosine of the sum of two angles. The angles x and y remaining fixed, we can imagine all of the lines to decrease indefinitely, and the functions sx, ex, sy, etc., are the limits of certain ratios of these lines.
Page 91 - ... often written that we will give only a brief outline. There is one axiom of Euclid that is somewhat complicated in its expression and does not seem to be, like the rest, a simple elementary fact. It is this : * If two lines are cut by a third, and the sum of the interior angles on the same side of the cutting line is less than two right angles, the lines will meet on that side when sufficiently produced. Attempts were made by many mathematicians, notably by Legendre, to give a proof of this proposition...
Page 8 - The sum of two lines drawn from a point to the extremities of a straight line is greater than the sum of two other lines similarly drawn, but included by them.
Page 32 - CPE, which is called the_ angle of parallelism for the perpendicular distance PC,. It is less than a right angle by an amount which is the limit of the deficiency of the triangle PCD. On the other side of PC we can find another line parallel to CA and making with PC the same angle of parallelism. We say that PE is parallel to AB towards that part which is on the same side of PC with PE. Thus, at any point there are two parallels to a line, but only one towards one part of the line. Lines through...
Page 52 - ... infinitesimal plane triangle, for which the sum of the angles is two right angles, and the angles and sides have the same relations as in the Euclidean Plane Geometry. All the formulae of Plane Trigonometry with which we are familiar hold, then, for triangles on the boundary-surface. On the boundary-surface we have the " hypothesis of the right angle.
Page 10 - The following propositions of Solid Geometry depend directly on the preceding and hold true at least for any restricted portion of space. 14. Theorem. If a line is perpendicular to two intersecting lines at their intersection, it is perpendicular to all lines of their plane passing through this point. 15. Theorem. If two planes are perpendicular, a line drawn in one perpendicular to their intersection is perpendicular to the other, and a line drawn through any point of one perpendicular to the other...