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KING。 I do not in the least understand you。
I。 Alas！ How shall I make it clear？ When you move straight on； does it not sometimes occur to you that you COULD move in some other way； turning your eye round so as to look in the direction towards which your side is now fronting？ In other words； instead of always moving in the direction of one of your extremities； do you never feel a desire to move in the direction； so to speak； of your side？
KING。 Never。 And what do you mean？ How can a man's inside 〃front〃 in any direction？ Or how can a man move in the direction of his inside？
I。 Well then； since words cannot explain the matter； I will try deeds； and will move gradually out of Lineland in the direction which I desire to indicate to you。
At the word I began to move my body out of Lineland。 As long as any part of me remained in his dominion and in his view； the King kept exclaiming； 〃I see you； I see you still； you are not moving。〃 But when I had at last moved myself out of his Line； he cried in his shrillest voice； 〃She is vanished； she is dead。〃 〃I am not dead；〃 replied I； 〃I am simply out of Lineland； that is to say； out of the Straight Line which you call Space； and in the true Space； where I can see things as they are。 And at this moment I can see your Line； or side or inside as you are pleased to call it； and I can see also the Men and Women on the North and South of you； whom I will now enumerate； describing their order； their size； and the interval between each。〃
When I had done this at great length； I cried triumphantly； 〃Does that at last convince you？〃 And； with that； I once more entered Lineland； taking up the same position as before。
But the Monarch replied； 〃If you were a Man of sense though； as you appear to have only one voice I have little doubt you are not a Man but a Woman but； if you had a particle of sense； you would listen to reason。 You ask me to believe that there is another Line besides that which my senses indicate； and another motion besides that of which I am daily conscious。 I； in return； ask you to describe in words or indicate by motion that other Line of which you speak。 Instead of moving； you merely exercise some magic art of vanishing and returning to sight； and instead of any lucid description of your new World； you simply tell me the numbers and sizes of some forty of my retinue； facts known to any child in my capital。 Can anything be more irrational or audacious？ Acknowledge your folly or depart from my dominions。〃
Furious at his perversity； and especially indignant that he professed to be ignorant of my sex； I retorted in no measured terms； 〃Besotted Being！ You think yourself the perfection of existence； while you are in reality the most imperfect and imbecile。 You profess to see； whereas you can see nothing but a Point！ You plume yourself on inferring the existence of a Straight Line； but I CAN SEE Straight Lines； and infer the existence of Angles； Triangles； Squares； Pentagons； Hexagons； and even Circles。 Why waste more words？ Suffice it that I am the completion of your incomplete self。 You are a Line； but I am a Line of Lines； called in my country a Square： and even I； infinitely superior though I am to you； am of little account among the great nobles of Flatland； whence I have come to visit you； in the hope of enlightening your ignorance。〃
Hearing these words the King advanced towards me with a menacing cry as if to pierce me through the diagonal； and in that same moment there arose from myriads of his subjects a multitudinous war…cry； increasing in vehemence till at last methought it rivalled the roar of an army of a hundred thousand Isosceles； and the artillery of a thousand Pentagons。 Spell…bound and motionless； I could neither speak nor move to avert the impending destruction； and still the noise grew louder； and the King came closer； when I awoke to find the breakfast…bell recalling me to the realities of Flatland。
Section 15。 Concerning a Stranger from Spaceland
From dreams I proceed to facts。
It was the last day of the 1999th year of our era。 The pattering of the rain had long ago announced nightfall； and I was sitting in the company of my wife； musing on the events of the past and the prospects of the coming year； the coming century； the coming Millennium。
＇Note： When I say 〃sitting〃； of course I do not mean any change of attitude such as you in Spaceland signify by that word； for as we have no feet； we can no more 〃sit〃 nor 〃stand〃 （in your sense of the word） than one of your soles or flounders。
Nevertheless； we perfectly well recognize the different mental states of volition implied in 〃lying〃； 〃sitting〃； and 〃standing〃； which are to some extent indicated to a beholder by a slight increase of lustre corresponding to the increase of volition。
But on this； and a thousand other kindred subjects； time forbids me to dwell。＇
My four Sons and two orphan Grandchildren had retired to their several apartments； and my wife alone remained with me to see the old Millennium out and the new one in。
I was rapt in thought； pondering in my mind some words that had casually issued from the mouth of my youngest Grandson； a most promising young Hexagon of unusual brilliancy and perfect angularity。 His uncles and I had been giving him his usual practical lesson in Sight Recognition； turning ourselves upon our centres； now rapidly； now more slowly； and questioning him as to our positions； and his answers had been so satisfactory that I had been induced to reward him by giving him a few hints on Arithmetic； as applied to Geometry。
Taking nine Squares； each an inch every way； I had put them together so as to 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 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 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 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 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 3； and if a straight Line of three inches； moving parallel to itself； makes a Square of three inches every 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 3^3。〃
〃Go to bed；〃 said I； a little ruffled by this interruption： 〃if you would talk less nonsense； you would remember more sense。〃
So my Grandson had disappeared in disgrace； and there I sat by my Wife's side； endeavouring to form a retrospect of the year 1999 and of the possibilities of the year 2000； but not quite able to shake off the thoughts suggested by the prattle of my bright little Hexagon。 Only a few sands now remained in the half…hour glass。 Rousing myself from my reverie I turned the glass Northward for the last time in the old Millennium； and in the act； I exclaimed aloud； 〃The boy is a fool。〃
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 round in every direction I could see nothing； yet still I FELT a Presence； and shivered