Page:Popular Science Monthly Volume 68.djvu/27

Rh Modern objections to these axioms are to the effect that most of them are too general to be true, that 2, 3, 4, 9, for example, are not valid in every case where we use the term equality; that the axioms are insufficient in that Euclid uses assumptions not explicitly stated, etc. But our present interest in looking for such faults is not great.

Of all the axioms and postulates, the last is by far the most remarkable and important historically. One is led from internal evidence to believe that Euclid introduced it only after failing to make his proofs without its aid. It is not used before proposition 29, not even in proposition 27 which states that if one line falls on two others

so as to make the 'alternate interior angles' (A and A ' ) equal, then the lines are parallel, i. e., do not meet. In proving the converse statement (29), however, he found it necessary to assume that if the sum of the two angles A '  and B is less than two right angles the lines will meet when produced far enough. This assumption is axiom 12.

It is perhaps worth while to add that the parallel axiom of which we are speaking may also be stated in the form: 'Through a point, A, in a plane, α, not more than one line can be drawn which does not intersect a line, a, lying in α but not itself passing through A.' The thirty-second proposition, to the effect that an exterior angle of a triangle is equal to the sum of the opposite interior angles, may also be used in place of axiom 12.

The twelfth axiom of Euclid was a stumbling block to many philosophers and mathematicians. While they were ready to grant that they would not be able to reason logically without the other axioms, this one seemed somehow less evident and less fundamental. The natural first attempt was to construct a proof for the axiom so as to give it place as a theorem. Many so-called demonstrations have been offered even up to the present day, but none that have withstood examination. At last, however, the thought came, "what if this axiom were not true? What would become of geometry if axiom 12 were replaced by a new axiom directly in contradiction with it?" It was found that by reasoning based on the reverse of axiom 12 one could involve himself in no contradiction, that, on the contrary, there