"The way that geometers... went about proving the fifth postulate was to smuggle in somewhere some unavowed assumption. A common practice was to assume that two straight lines could not approach one another assymptotically, that... they ultimately intersected. Or, again, it was assumed that a straight line was not a closed circuit... legitimate as long as avowed. A franker, and so more admirable way... was to change the definition of parallel lines into something else that seemed to avoid the trouble, or else to reword the axiom in a less objectionable form. A real step in advance... is known as Playfair's axiom, though it is casually mentioned in Proclus...There are... a great many alternatives. One of the most famous is to define two coplaner lines as parallel if they are everywhere the same distance apart... but how do we know there are such pairs... A still neater method consists in defining two lines as parallel if they have the same direction, or opposite directions. But here we introduce a totally new undefined concept, direction..."