Men of Mathematics by E.T. Bell

Author:E.T. Bell
Language: eng
Format: epub
Publisher: Touchstone

There are one or two simple matters to be disposed of before we come to Lobatchewsky’s Copernican part in the extension of geometry. We have alluded to “equivalents” of the parallel postulate. One of these, “the hypothesis of the right angle,” as it is called, will suggest two possibilities, neither equivalent to Euclid’s assumption, one of which introduces Lobatchewsky’s geometry, the other, Riemann’s.

Consider a figure AXTB which “looks like” a rectangle, consisting of four straight lines AX, XT, TB, BA, in which BA (or AB)is the base, AX and TB (or BT) are drawn equal and perpendicular to AB, and on the same side of AB. The essential things to be remembered about this figure are that each of the angles XA B, TBA (at the base) is a right angle, and that the sides AX, BY are equal in length. Without using the parallel postulate, it can be proved that the angles AXT, BTX, are equal, but, without using this postulate, it is impossible to prove that AXT, BTX are right angles, although they look it. If we assume the parallel postulate we can prove that AXT, BTX are right angles and, conversely, if we assume that AXT, BTX are right angles, we can prove the parallel postulate. Thus the assumption that AXT, BTX are right angles is equivalent to the parallel postulate. This assumption is today called the hypothesis of the right angle (since both angles are right angles the singular instead of the plural “angles” is used).

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