Monographs on Topics of Modern Mathematics, Relevant to the Elementary FieldJacob William Albert Young |
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Common terms and phrases
algebraic equation angle of parallelism anharmonic ratio anti-derivative Assumption called coefficients complete quadrangle complex quantities congruent conic convex regions Corollary curve defined definition degree denoted distance domain of rationality elementary elements equal Euclid Euclidean Euclidean geometry example expressed fact factor finite number follows formula fundamental circle given gruent harmonic Hence imaginary imaginary units infinitely distant integers integral function interval irrational lines joining Lobachevskian mathematical method multiple non-Euclidean geometries nth roots pairs of corresponding parallel perpendicular polar polynomial positive integer prime proof properties propositions proved quadratic equation quadrilateral rational functions rational numbers real numbers real points respectively Riemannian geometry right angles roots of unity satisfy sheaf of rays sheaves sides solution straight line substitutions symbol tangent Theorem theory of equations tion uniquely determined unknowns values vertices zero
Popular passages
Page 93 - 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.
Page 83 - The points of intersection of the three pairs of opposite sides of a hexagon inscribed in a conic lie on one straight line.
Page 41 - If one side of a triangle be produced, the exterior angle is greater than either of the interior, and opposite angles.
Page 259 - A sufficient condition for the maximum number of imaginary roots of an equation of the nth degree,
Page 354 - A proposed construction is possible by ruler and compasses if, and only if, the numbers which define analytically the desired geometric elements can be derived from those defining the given elements by a finite number of rational operations and extractions of real square roots.
Page 32 - SSS); two sides and the included angle of one triangle are congruent to the corresponding parts...
Page 391 - The third period extends from the middle of the eighteenth century to the present time...
Page 362 - Let a real number which can be obtained from the integers by a finite number of rational operations and extractions of square roots be called a quadratic number. A, B, C are any three points not in a straight line such that AC and BC are quadratic in terms of AB.
Page 195 - This system satisfies all the postulates except Postulate 27. It is larger than the system of ordinary complex quantities, and contains that system just as the system of ordinary complex quantities contains the system of real quantities. Postulate 27 is therefore a restrictive condition. 36. What is algebra? We are now in a position to answer the question, " What is the algebra of complex quantities?
Page 184 - All real points which can be expressed in the form ±m/n, where m and n are any positive integral points [sec. 25, (17)] together with the point 0, are called the rational points. The rational points which are not integral are called fractional; the fractional points lie between the integral points. All real points which are not rational are called irrational. That not all the real points are "rational...