Page:Carroll - Euclid and His Modern Rivals.djvu/287

Rh be halved and quartered, is the 'difference of the direction' of 'two straight lines that meet one another.' A better definition follows; the 'quantity of turning' by which we pass from one direction to another. But hardly any use is made of this, and none at the commencement. And why two definitions? Is the difference of two directions the same thing as the rotation by which we pass from one to the other? Is the difference of position of London and Rugby a number of miles on the railroad? Yes, in a loosely-derived and popular and slip-slop sense: and in like manner we say that one man is a pigeon-pie, and another is a shoulder of lamb, when we describe their contributions to a pic-nic. But non est geometria! Metaphor and paronomasia can draw the car of poetry; but they tumble the waggon of geometry into the ditch.

Parallels, of course, are lines which have the same direction. It is stated, as an immediate consequence, that two lines which meet cannot make the same angle with a third line, on the same side, for they are in different directions. Parallels are knocked over in a trice. There is a covert notion of direction, which, though only defined with reference to lines which meet, is straightway transferred to lines which do not. According to the definition, direction is a relation of lines which do meet, and yet lines which have the same direction can be lines which never meet. There is a great quantity of turning wanted; turning of implied assumption into expressed. Mr. Wilson would, we have no doubt, immediately introduce and defend all we ask for; and we quite admit that his system has a right to it. How do you know, we ask, that lines which have the same direction never meet? Answer—lines which meet have different directions. We know they have; but how do we know that, under the definition given, the relation called direction has any application at all to lines which never meet? The use of the notion of limits may give an answer: but what is the system of geometry which introduces continuity and limits to the mind as yet untaught to think of space and of