Page:Popular Science Monthly Volume 67.djvu/647

Rh He then proceeds to misquote it as follows:

Si une droite, en coupant deux autres droites, fait les angles internes inégaux, ou moindres que deux angles droits, ces deux droits, prolongées à rinfini, se rencontreront du côté ou les angles sont plus petits que deux droits;

and continues,

It is certain that, placed after the definitions, this Postulatum is incomprehensible. But, placed after Proposition XXVI. of the first book, where the author demonstrates that 'if the interior angles together equal two right angles, the lines will not meet,' it acquires almost the evidence of an axiom.

The XXVI. is, of course, a mistake for XXVIII.

Other mathematicians have tried to turn the flank of the difficulty by substituting a new definition of parallels for Euclid's.

Eu. I., Def. 35, is: 'Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet.'

On this Hobbs petulantly remarks:

How shall a man know that there be straight lines which shall never meet, though both ways infinitely produced?

The answer is simple: Read Eu. I., 27, where if the straight line be infinite, is proven that those making equal alternate angles nowhere meet.

Wolf, Boscovich and T. Simpson substitute for Euclid's the definition: 'Straight lines are parallel which preserve always the same distance from each other.' But this is begging the question, since it assumes the definition, 'two straight lines are parallel when there are two points in the one on the same side of the other from which the perpendiculars to it are equal,' and at the same time assumes the theorem, 'all perpendiculars from one of these lines to the other are equal.'

Just so the assumption that there are straights having the same direction is a petitio principii, since it assumes the definition of Varignon and Bézout, that 'parallel lines are those which make equal angles with a third line,' and at the same time assumes the theorem that 'straight lines which make equal angles with one given transversal make equal angles with all transversals.'

Other and more penetrating geometers have proposed substitutes for the parallel-postulate. Of these the simplest are Ludlam's: 'Two straight lines which cut one another can not both be parallel to the same straight line,' and W. Bolyai's 'Any three points are costraight or concyclic.'

But the largest and most desperate class of attempts to remove this supposed blemish from geometry consists of those who strive to deduce the theory of parallels from reasonings about the nature of the straight line and plane angle, helped out by Euclid's nine other assumptions