Page:Carroll - Euclid and His Modern Rivals.djvu/144

106 'the same pair of directions'; and if they be said to have 'different directions,' we must understand 'different pairs of directions.' And even this is not enough: for suppose I draw, on the map of England, a straight Line joining London and York; I may say 'This Line has a pair of directions, the first being "London-to-York" and the second "York-to-London. I will now place another Line upon this, and its pair of directions shall be, first "York-to- London" and second "London-to-York." Then it has a different first-direction from the former Line, and also a different second-direction: that is, it has a 'different pair of directions.' Clearly this is not intended: but, in order to exclude such a possibility, we must extend yet further the meaning of the phrase, and, if two Lines be said to have 'the same direction,' we must understand 'pairs of directions which can be arranged so as to be the same'; and if they be said to have 'different directions,' we must understand 'pairs of directions which cannot be arranged so as to be the same.' Nie. Yes, that expresses our meaning.

Min. You must admit, I think, that your theory of direction involves a good deal of obscurity at the very outset. However, we have cleared it up, and will not use the word 'opposite' again. Tell me now whether you accept this as a correct Definition of the phrases 'the same direction' and 'different directions,' when used of a Pair of infinite Lines which have a common point:—

If two infinite Lines, having a common point, coincide, they have 'the same direction'; if not, they have 'different directions.'