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 8 - If two triangles have two sides of the one equal respectively to two sides of the other, but the third side of the first greater than the third side of the second, then the included angle of the first is greater than the included angle of the second. [Converse of Prop. XXXI.] B' b' Given A ABC and A'B'C', with b = b'; c = c'; a > a . To prove Z A> Z A'.
Page 9 - ... 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 line determine a perpendicular to the line at its middle point. 10, Theorem. Two triangles...
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 - 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 ivhen sufficiently produced.
Page 91 - ... sufficiently produced. 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 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 52 - On the boundary-surface we have the " hypothesis of the right angle." Rectangles can be formed, and the area of a rectangle is proportional to the product of its base and altitude, while the area of a triangle is half of the area of a rectangle having the same base and altitude.

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