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This backward journey must again stop somewhere so that there are some propositions which are accepted as true and for which proof is not required. 9. Are the unproved propositions or postulates subject to no restrictions at all ? They are subject to two important restrictions. The first is that the postulates must be consistent. This means that contradictory statements are not implied by the postulates. They must not lead to 'A is B' and 'A is not B'. The second is that the postulates must be complete.

It would appear that an infinite number of geometries is possible, because starting with any postulates whatever a geometry can be built up, provided only that the postulates do not lead to any contradiction. A surface can often be found to which a new geometry will be applicable. Alternatively, however complex a surface may be, a geometry peculiarly suited to it could be constructed. 51. What is Projective Geometry ? Consider an artist standing before a scene he wants to paint. His canvas can be imagined to be a glass screen intercepted between the seene and his eye.

But a few mathematicians, lobachewsky being one of them, thought of seeing what happened if the postulate were replaced by the following one: " Through a given point which is not on a given straight line, two different lines can be drawn both of which are parallel to the given line". We could try to draw a figure like the following, where two distinct lines are drawn through the point P, one to the right and the other to the left. p • A D • B Mathematicians found that not only no fallacy resulted from this strange assumption but a logical consequence of the new assumption 6 Geometry and Geometries led them to a new geometry in which the sum of the angles of a triangle is less than 1800 • 22.