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Moritz Cantor, the historian of mathematics

1920; American Mathematical Society; Volume: 27; Issue: 1 Linguagem: Inglês

10.1090/s0002-9904-1920-03351-3

1088-9485

Florian Cajori,

Philosophy and History of Science

side.Let the side z±z 2 be the #-axis.The equation of the tangent parallel to this axis iswith similar equations of the tangents parallel to the other two sides.Let / X3 be negative.The points of contact with the sides z\z z and z 2 z z lie between the #-axis and the parallel tangent or outside them according as Mi + M2 + M3 > 0, and the conic is accordingly an ellipse or hyperbola.Hence if all the powers be positive <p = 0 is an ellipse.If one of the powers be negative it is an ellipse, parabola or hyperbola according as their sum is less than, equal to or greater than zero.If two of the powers be negative it is an ellipse, parabola or hyperbola according as their sum is greater than, equal to or less than zero.In particular if Mi + M3 = 0 the side ZiZ Z is an asymptote.If fxi = \x 2 = -M3 the sides z\Zz and z 2 z z are asymptotes.If in addition the triangle z\z 2 z z is isosceles having its vertex at z Zi the circle having zs for its center and passing through zi and z 2 passes through the foci of <p = 0, for the altitude and half base of this triangle are the major and minor axes respectively.