Question

State True or False.
Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.
A. TRUE
B. FALSE

Answer 1

Hint: Here we tell how Euclid’s postulates and axioms helped to prove Euclid’s fifth postulate and check if other geometries were discovered or not.

Complete step-by-step answer:
Euclid was a Greek mathematician who was called founder of geometry. Euclid stated the fifth postulate without proving it and many attempts were made to prove it but failed. Euclid’s fifth postulate states 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 two right angles.”
Attempts to prove this postulate were made by:
1) Earliest attempt was made by Proclus in the fifth century. The commentary on Euclid’s elements is unlikely to be complete, he left a proof of his own but the latter assumes that parallel lines are always bounded distance apart and this can be used to prove fifth postulate.
2) Al-Gauhary made an attempt to prove the fifth postulate in the ninth century. He assumed that if alternating angles determined by a line cutting two lines are equal then the same will be true for all lines.
3) Al-Haytham’s made an attempt to prove the postulate in the tenth century. His method was criticized by Omar Khayyam in the eleventh century. Later in thirteen century Nasir ad-Din at-Tusi analyzed the work of Al-Haytham, Omar Khayyam and Al-Gauhary. He tried to prove postulate using contradiction by assuming the fifth postulate is wrong.
4) John Wallis in 1633 tried proving the postulate by assuming that there is a similar figure of arbitrary size for any figure.
We can see all the mathematicians who attempted to prove Euclid's fifth postulate in a way made a new geometrical statement true.
SO the answer to the statement in question is TRUE.


So, the correct answer is “Option A”.

Note: Students should always focus on all the attempts made as some do not give any output whereas some gave new discoveries regarding geometry.