Page:Carroll - Euclid and His Modern Rivals.djvu/154

116 different directions, you ask me to believe that they will meet if produced?

Nie. That is our meaning.

Min. We had better heave the lead once more, and return to our fixed Line, and a point not on it, through which we wish to draw a second Line. You ask me to grant that, if it be drawn so as to have a direction 'different' from that of the first Line, it will meet it if prolonged indefinitely?

Nie. That is our humble petition.

Min. Will you be satisfied if I grant you that some Lines, so drawn, will meet the first Line? That I would grant you with pleasure. I could draw millions of Lines which would fulfil the conditions, by simply taking points at random on the given Line, and joining them to the given point. Every Line, so constructed, would have a direction 'different' from that of the given Line, and would also meet it.

Nie. We will not be satisfied, even with millions! We ask you to grant that every Line, drawn through the given point with a direction 'different' from that of the given Line, will meet the given Line: and we ask you to grant this independently of, and antecedently to, any other information about the Lines except the fact that they have 'different' directions.

Min. But what meaning am I to attach to the phrase 'different directions,' independently of, and antecedently to, the fact that they have a common point?

Nie. (after a long silence) I fear we can suggest none.

Min. Then I must decline to accept the Axiom.