Studies in Non-Euclidian Geometry

University of Wisconsin--Madison, 1925 - 100 sider

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Side 1 - 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.
Side 47 - The sum of the angles in a triangle is less than 180 degrees, the more so, the larger the sides of the triangle.
Side 43 - If two parallel lines are cut by a transversal, the corresponding angles are equal.
Side 2 - Euclid's twelfth axiom, which is more properly speaking a postulate, was his starting point for proving that through a given point one and only one line can be drawn parallel to a given line.
Side 39 - Ex. 11. Two lines of unequal length bisect each other at right angles. Show that any point in either line is equidistant from the extremities of the other. (§ 54.) PROP. VII. THEOREM 59. From a given point without a straight line, but one perpendicular can be drawn to the line. (It follows from § 25 that, from a given point without a straight line...
Side 45 - ... of geometry (I do not speak of those of arithmetic) are merely disguised definitions. Then what are we to think of that question : Is the Euclidean geometry true? It has no meaning. As well ask whether the metric system is true and the old measures false ; whether Cartesian coordinates are true and polar coordinates false. One geometry can not be more true than another ; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient...
Side 2 - The Hypothesis of the Right Angle. (2) The Hypothesis of the Obtuse Angle. (3) The Hypothesis of the Acute Angle.
Side 43 - Let a and b be the two v given lines, which neither intersect nor are parallel. From any two points A and P on the line a, draw AB and PB' perpendicular to the line 6. If AB = PB', the existence of a common perpendicular follows from § 28. Therefore we need only discuss the case when AB is...
Side 19 - The additive property, .!•£•» distance ab + distance be *= distance ac. (2) The distance from a point to Itself is zero. (3) The distance between two points is unaltered by а translation of the line...
Side 26 - A circle is a conic which has double contact with the absolute circle, whose axis IB the chord of contact and whose center is the pole of the axis with respect to the absolute or the circle Itself.

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