EUCLIDIAN AND NON EUCLIDIAN GEOMETERY (FORTH EDITION) -Find the flaw in the foll
ID: 2899700 • Letter: E
Question
EUCLIDIAN AND NON EUCLIDIAN GEOMETERY (FORTH EDITION) -Find the flaw in the following attempted proof of the parallel postulate given by J. D. Gergonne (see Figure 5.12). Given P not on line L, PQ perpendicular to L at Q, line m perpendicular to PQ at P, and point A not = P on m. Let PB be the last ray between PA and PQ that intersects L, B being the point of intersection. There exists a point C on L such that Q * B* C (Axioms B-1 and B-2). It follows that PB is not the last ray between PA and PQ that intersects L, and hence all rays between PA and PQ meet L. Thus m is the only parallel to L through p.
Explanation / Answer
PBPB is not the last ray between rays PAPA and PQPQ that intersects ll, and hence all rays between PAPAand PQPQ meet ll"
The "hence" here is asserted without proof. In fact, it does not follow. All we have shown is that given any point BB there is a point CC beyond it.
For example, given any real number bb less than 2, there is another real number less than 2 but greater than bb. Does it follow that all real numbers are less than 2?
Similarly, given any ray through PP intersecting ll at BB, we have shown there is another ray through PP intersecting ll at CC. Logically, it does not follow that all rays therefore intersect ll.
There's not need for a counterexample; we just need to show the reasoning is flawed.
Like many of these attempted proofs that popped up throughout history, it's just a clever way to subtly hide the assumption of the parallel postulate by cloaking it in "geometric intuition". The "hence" is an appeal to the imagination; it is hard to imagine that such a ray between PAPA and PQPQcan exist without intersecting ll, but that's not proof that it doesn't exist.
Edit: Ironically, people are downvoting my answer because they fell into the same trap Gergonne fell into. These would probably be the people (many of whom were otherwise skilled mathematicians) who accepted Gergonne's proof back in the day. What Gergonne proves by contradiction is that there is no last ray between PAPA and PQPQ that intersects ll. That is not the same as saying there is no ray between PAPA and PQPQ that does not intersect l. They are not the same thing. With a little more structure you can think of it like this: he proves that the set of all the angles between PQPQ and the rays that intersect ll has no greatest element. Logically it's still possible that it's bounded from above by some angle less than a right angle, so he hasn't proved what he set out to prove.
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