When my Grandson entered the room I carefully secured the door. Then, sitting down by his side and taking our mathematical tablets,—or, as you would call them, Lines—I told him we would resume the lesson of yesterday. I taught him once more how a Point by motion in One Dimension produces a Line, and how a straight Line in Two Dimensions produces a Square. After this, forcing a laugh, I said, “And now, you scamp, you wanted to make believe that a Square may in the same way by motion ‘Upward, not Northward’ produce another figure, a sort of extra square in Three Dimensions. Say that again, you young rascal.” At this moment we heard once more the herald’s “O yes! O yes!” outside in the street proclaiming the Resolution of the Council. Young though he was, my Grandson—who was unusually intelligent for his age, and bred up in perfect reverence for the authority of the Circles—took in the situation with an acuteness for which I was quite unprepared. He remained silent till the last words of the Proclamation had died away, and then, bursting into tears, “Dear Grandpapa,” he said, “that was only my fun, and of course I meant nothing at all by it; and we did not know anything then about the new Law; and I don’t think I said anything about the Third Dimension; and I am sure I did not say one word about ‘Upward, not Northward,’ for that would be such nonsense, you know. How could a thing move Upward, and not Northward? Upward and not Northward! Even if I were a baby, I could not be so absurd as that. How silly it is! Ha! ha! ha!” “Not at all silly,” said I, losing my temper; “here for example, I take this Square,” and, at the word, I grasped a moveable Square, which was lying at hand—“and I move it, you see, not Northward but—yes, I move it Upward—that is to say, Northward but I move it somewhere—not exactly like this, but somehow—” Here I brought my sentence to an inane conclusion, shaking the Square about in a purposeless manner, much to the amusement of my Grandson, who burst out laughing louder than ever, and declared that I was not teaching him, but joking with him; and so saying he unlocked the door and ran out of the room. Thus ended my first attempt to convert a pupil to the Gospel of Three Dimensions. § 22 How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result My failure with my Grandson did not encourage me to communicate my secret to others of my household; yet neither was I led by it to despair of success. Only I saw that I must not wholly rely on the catch-phrase, “Upward, not Northward,” but must rather endeavour to seek a demonstration by setting before the public a clear view of the whole subject; and for this purpose it seemed necessary to resort to writing.