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addition and multiplication axioms bounding sphere called cardinal number characterize circle class of points commutative complex numbers conception concrete representation congruent consider consistent defined definition denumerable discrete sequence discussion E. H. Moore elementary equal equation equivalent Euclid's Euclid's Elements euclidean geometry example fact fifth postulate finite follows formal logical four-dimensional fractions function fundamental geometric interpretation infinite number interval irrational numbers last element limit linear linear order m-class containing mathe mathematical science means method metric geometry negative numbers non-euclidean geometry notion number of elements number system obtained one-to-one correspondence parallel parallel postulate point of view positive integers positive number problem projective geometry proof properties pupil quaternions question rational numbers readily seen real numbers regarded relation represented satisfied segment set of assumptions shortest lines space sphere square straight line Suppose symbol term mathematical theorem tion triangle types of order undefined terms variable zero
Page 26 - ... far prolonged, the same distance apart, that is, never intersect. They have the properties of the Euclidean parallels, and may be called and defined as such. It likewise follows, now, from the properties of triangles and rectangles, that two straight lines which are cut by a third straight line so as to make the sum of the interior angles on the same side of them less than two right angles will meet on that side, but in either direction from their point of intersection will move indefinitely...
Page 157 - E is a point on CA distinct from C and A, then there is a point F on AB such that D, E, F are collinear.
Page 222 - A mathematical science is any body of propositions which is capable of an abstract formulation and arrangement in such a way that every proposition of the set after a certain one is a formal logical consequence of some or all the preceding propositions. Mathematics consists of all such mathematical sciences.
Page 21 - Method.) 170. Parallel lines are lines which lie in the same plane and do not meet, however far they are produced.
Page 34 - Riemann's geometry, the sum of the angles of a triangle is always greater than two right angles.
Page 172 - It may not be out of place at this point to say a word regarding the conception of a space of four or more dimensions.
Page 11 - 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 53 - Pure Mathematics is the class of all propositions of the form "p implies q," where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants.