THE FOUNDATIONS OF GEOMETRY 23 



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 re- 

 markable and important historically. One is led from internal evi- 

 dence 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 



Fig. 1. Fig. 2. 



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, a, not more than one line can be drawn which does not 

 intersect a line, a, lying in a 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 philos- 

 ophers 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 ex- 

 amination. 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 re- 



