

@ 2147,27 +2147,27 @@ make one large Square, with a side of three inches, and I had hence proved to 



my little Grandson that  though it was impossible for us to see the inside of 



the Square  yet we might ascertain the number of square inches in a Square by 



simply squaring the number of inches in the side: ``and thus,'' said I, ``we know 



that threetothesecond, or nine, represents the number of square inches in a 



Square whose side is three inches long.'' 



that $ 3^2 $, or 9, represents the number of square inches in a 



Square whose side is 3 inches long.'' 







The little Hexagon meditated on this a while and then said to me; ``But you 



have been teaching me to raise numbers to the third power: I suppose 



threetothethird must mean something in Geometry; what does it mean?'' 



$ 3^3 $ must mean something in Geometry; what does it mean?'' 



``Nothing at all,'' replied I, ``not at least in Geometry; for Geometry has only 



Two Dimensions.'' And then I began to shew the boy how a Point by moving 



through a length of three inches makes a Line of three inches, which may be 



represented by three; and how a Line of three inches, moving parallel to 



represented by 3; and how a Line of three inches, moving parallel to 



itself through a length of three inches, makes a Square of three inches every 



way, which may be represented by threetothesecond. 



way, which may be represented by $ 3^2 $. 







Upon this, my Grandson, again returning to his former suggestion, took me up 



rather suddenly and exclaimed, ``Well, then, if a Point by moving three inches, 



makes a Line of three inches represented by three; and if a straight Line of 



makes a Line of three inches represented by 3; and if a straight Line of 



three inches, moving parallel to itself, makes a Square of three inches every 



way, represented by threetothesecond; it must be that a Square of three 



way, represented by $ 3^2 $; it must be that a Square of three 



inches every way, moving somehow parallel to itself (but I don't see how) must 



make Something else (but I don't see what) of three inches every way  and 



this must be represented by threetothethird.'' 



this must be represented by $ 3^3 $.'' 







``Go to bed,'' said I, a little ruffled by this interruption: ``if you would talk 



less nonsense, you would remember more sense.'' 


@ 2184,12 +2184,12 @@ Straightway I became conscious of a Presence in the room, and a chilling 



breath thrilled through my very being. ``He is no such thing,'' cried my Wife, 



``and you are breaking the Commandments in thus dishonouring your own 



Grandson.'' But I took no notice of her. Looking around in every direction I 



could see nothing; yet still I felt a Presence, and shivered as the cold 



could see nothing; yet still I \textit{felt} a Presence, and shivered as the cold 



whisper came again. I started up. ``What is the matter?'' said my Wife, ``there 



is no draught; what are you looking for? There is nothing.'' There was nothing; 



and I resumed my seat, again exclaiming, ``The boy is a fool, I say; 



threetothethird can have no meaning in Geometry.'' At once there came a 



distinctly audible reply, ``The boy is not a fool; and threetothethird has 



$ 3^3 $ can have no meaning in Geometry.'' At once there came a 



distinctly audible reply, ``The boy is not a fool; and $ 3^3 $ has 



an obvious Geometrical meaning.'' 







My Wife as well as myself heard the words, although she did not understand 


