Page:Carroll - Euclid and His Modern Rivals.djvu/146

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P. 11. Ax. 6. 'Two different Lines may have either the same or different directions.'

Min. That contains two assertions, which we will consider separately. First, you say that 'two different Lines (i.e. 'non-coincidental Lines,' or 'Lines having a separate point') 'may have the same direction'. Now let us understand each other quite clearly. We will take a fixed Line to begin with, and a certain point on it: there is no doubt that we can draw, through that point, a second Line coinciding with the first: the direction of this Line will of course be 'the same' as the direction of the first Line; and it is equally obvious that if we draw the second Line in any other direction, so as not to coincide with the first, its direction will not be 'the same' as that of the first: that is, they will have 'different' directions. If we want a geometrical definition of the assertion that this second Line has 'the same direction' as the first Line, we may take the following:—'having such a direction as will cause the Lines to be the same Line.' If we want a geometrical construction for it, we may say 'take any other point on the fixed Line; join the two points, and produce the Line, so drawn, at both ends': this construction we know will produce a Line which will be 'the same' as the first Line, and whose direction will therefore be 'the same' as that of the first Line. If, in a certain diagram, whose geometrical history we know, we want to test whether two Lines, passing through a common point, have, or have not, 'the same direction,' we have simply to take