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algebra 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 exists fact fifth postulate finite follows formal logical four-dimensional fractions function fundamental geometric interpretation infinite number interval irrational numbers last element last lecture 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 plane 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 theorem tion triangle types of order undefined terms variable zero
Page 53 - Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Page 107 - the second value is in this case not to be taken, for it is inadequate ; people do not approve of negative roots.
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 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 - 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 32 - Another point which we may emphasize at this time is the fact that the set of assumptions which they made are thoroughly self -consistent ; starting with those assumptions, we could not expect in any way ever to arrive at a contradiction, and it is therefore evident that Euclid's postulate cannot be derived from the other postulates. The problem is now on a par with the squaring of the circle and the trisection of an angle by means of ruler and compass. So far as the mathematical public is concerned,...
Page 182 - ... branches" is entirely wiped out. That this is so is readily seen. We saw in the last lecture how it is possible to represent the points of space by sets of three coordinates. It is therefore possible, starting with the set of assumptions characterizing the algebra of real numbers, to define a system of things which is abstractly equivalent to metric euclidean geometry. We merely define a "point...
Page 236 - And to avoide the tediouse repitition of these woordes : is equalle to: I will sette as I doe often in woorke use, a paire of parallelles, or Gemowe lines of one lengthe, thus : ~ , bicause noe. 2. thynges can be moare equalle.