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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 rotation 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 9 - Notions 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another.
Page 105 - the second value is in this case not to be taken, for it is inadequate ; people do not approve of negative roots.
Page 24 - ... 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 220 - 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 155 - 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 19 - Proposition XIV may now be stated : When two straight lines are cut by a transversal, if the alternate-interior angles are equal, the straight lines are parallel. The figures below show that if the alternate-interior angles...
Page 218 - It seems that with reference to the classes ordinarily considered in mathematics all we can say is that, in spite of the enormous amount of work that has been done on them, no contradiction has ever appeared.
Page 32 - Riemann's geometry, the sum of the angles of a triangle is always greater than two right angles.
Page 17 - ... the object will travel along the arc PQ and reach the eye of the man by the shortest path. If he walked toward the object, keeping it always directly in view, he would move along the arc PQ and arrive at Q by the smallest possible number of steps. We see, then, that these shortest lines play a rdle in his geometry very similar to that which straight lines play in ours; in fact, these shortest lines would "look straight