Non-Euclidean Geometry

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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 - 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 line determine a perpendicular to the line at its middle point. 10. Theorem. Two triangles are equal when they have three sides of one equal, respectively, to three sides of the other. 11. Theorem. If two lines in a plane erected perpendicular...
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 91 - 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.
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 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 91 - Attempts were made by many mathematicians, notably by Legendre, to give a proof of this proposition ; that is, to show that it is a necessary consequence of the simpler axioms preceding it. Legendre proved that the sum of the angles of a triangle can never exceed two right angles, and that if there is a single triangle in which this sum is equal to two right angles, the same is true of all triangles. This was, of course, on the supposition that a line is of infinite length.
Page 32 - 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 P which make with PC an angle greater than the angle of parallelism and less than its...

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