

@ 2262,54 +2262,54 @@ humiliation that I, a Square, should have been guilty of the impertinence of 



feeling a Circle. It was commenced by the Stranger with some impatience at the 



lengthiness of my introductory process. 







STRANGER. Have you felt me enough by this time? Are you not introduced to me 



\textit{Stranger}. Have you felt me enough by this time? Are you not introduced to me 



yet? 







I. Most illustrious Sir, excuse my awkwardness, which arises not from 



\textit{I}. Most illustrious Sir, excuse my awkwardness, which arises not from 



ignorance of the usages of polite society, but from a little surprise and 



nervousness, consequent on this somewhat unexpected visit. And I beseech you 



to reveal my indiscretion to no one, and especially not to my Wife. But before 



your Lordship enters into further communications, would he deign to satisfy 



the curiosity of one who would gladly know whence his visitor came? 







STRANGER. From Space, from Space, Sir: whence else? 



\textit{Stranger}. From Space, from Space, Sir: whence else? 







I. Pardon me, my Lord, but is not your Lordship already in Space, your 



\textit{I}. Pardon me, my Lord, but is not your Lordship already in Space, your 



Lordship and his humble servant, even at this moment? 







STRANGER. Pooh! what do you know of Space? Define Space. 



\textit{Stranger}. Pooh! what do you know of Space? Define Space. 







I. Space, my Lord, is height and breadth indefinitely prolonged. 



\textit{I}. Space, my Lord, is height and breadth indefinitely prolonged. 







STRANGER. Exactly: you see you do not even know what Space is. You think it is 



\textit{Stranger}. Exactly: you see you do not even know what Space is. You think it is 



of Two Dimensions only; but I have come to announce to you a Third  height, 



breadth, and length. 







I. Your Lordship is pleased to be merry. We also speak of length and height, 



\textit{I}. Your Lordship is pleased to be merry. We also speak of length and height, 



or breadth and thickness, thus denoting Two Dimensions by four names. 







STRANGER. But I mean not only three names, but Three Dimensions. 



\textit{Stranger}. But I mean not only three names, but Three Dimensions. 







I. Would your Lordship indicate or explain to me in what direction is the 



\textit{I}. Would your Lordship indicate or explain to me in what direction is the 



Third Dimension, unknown to me? 







STRANGER. I came from it. It is up above and down below. 



\textit{Stranger}. I came from it. It is up above and down below. 







I. My Lord means seemingly that it is Northward and Southward. 



\textit{I}. My Lord means seemingly that it is Northward and Southward. 







STRANGER. I mean nothing of the kind. I mean a direction in which you cannot 



\textit{Stranger}. I mean nothing of the kind. I mean a direction in which you cannot 



look, because you have no eye in your side. 







I. Pardon me, my Lord, a moment's inspection will convince your Lordship that 



\textit{I}. Pardon me, my Lord, a moment's inspection will convince your Lordship that 



I have a perfectly luminary at the juncture of my two sides. 







Stranger: Yes: but in order to see into Space you ought to have an eye, not on 



\textit{Stranger}: Yes: but in order to see into Space you ought to have an eye, not on 



your Perimeter, but on your side, that is, on what you would probably call 



your inside; but we in Spaceland should call it your side. 







I. An eye in my inside! An eye in my stomach! Your Lordship jests. 



\textit{I}. An eye in my inside! An eye in my stomach! Your Lordship jests. 







STRANGER. I am in no jesting humour. I tell you that I come from Space, or, 



\textit{Stranger}. I am in no jesting humour. I tell you that I come from Space, or, 



since you will not understand what Space means, from the Land of Three 



Dimensions whence I but lately looked down upon your Plane which you call 



Space forsooth. From that position of advantage I discerned all that you speak 


@ 2317,9 +2317,9 @@ of as solid (by which you mean ``enclosed on four sides''), your houses, your 



churches, your very chests and safes, yes even your insides and stomachs, all 



lying open and exposed to my view. 







I. Such assertions are easily made, my Lord. 



\textit{I}. Such assertions are easily made, my Lord. 







STRANGER. But not easily proved, you mean. But I mean to prove mine. 



\textit{Stranger}. But not easily proved, you mean. But I mean to prove mine. 







When I descended here, I saw your four Sons, the Pentagons, each in his 



apartment, and your two Grandsons the Hexagons; I saw your youngest Hexagon 


@ 2328,22 +2328,22 @@ alone. I saw your Isosceles servants, three in number, in the kitchen at 



supper, and the little Page in the scullery. Then I came here, and how do you 



think I came? 







I. Through the roof, I suppose. 



\textit{I}. Through the roof, I suppose. 







STRANGER. Not so. Your roof, as you know very well, has been recently 



\textit{Stranger}. Not so. Your roof, as you know very well, has been recently 



repaired, and has no aperture by which even a Woman could penetrate. I tell 



you I come from Space. Are you not convinced by what I have told you of your 



children and household? 







I. Your Lordship must be aware that such facts touching the belongings of his 



\textit{I}. Your Lordship must be aware that such facts touching the belongings of his 



humble servant might be easily ascertained by any one of the neighbourhood 



possessing your Lordship's ample means of information. 







STRANGER. (To himself.) What must I do? Stay; one more argument suggests 



\textit{Stranger}. (To himself.) What must I do? Stay; one more argument suggests 



itself to me. When you see a Straight Line  your wife, for example  how many 



Dimensions do you attribute to her? 







I. Your Lordship would treat me as if I were one of the vulgar who, being 



\textit{I}. Your Lordship would treat me as if I were one of the vulgar who, being 



ignorant of Mathematics, suppose that a Woman is really a Straight Line, and 



only of One Dimension. No, no, my Lord; we Squares are better advised, and are 



as well aware of your Lordship that a Woman, though popularly called a 


@ 2351,35 +2351,35 @@ Straight Line, is, really and scientifically, a very thin Parallelogram, 



possessing Two Dimensions, like the rest of us, viz., length and breadth (or 



thickness). 







STRANGER. But the very fact that a Line is visible implies that it possesses 



\textit{Stranger}. But the very fact that a Line is visible implies that it possesses 



yet another Dimension. 







I. My Lord, I have just acknowledge that a Woman is broad as well as long. We 



\textit{I}. My Lord, I have just acknowledge that a Woman is broad as well as long. We 



see her length, we infer her breadth; which, though very slight, is capable of 



measurement. 







STRANGER. You do not understand me. I mean that when you see a Woman, you 



\textit{Stranger}. You do not understand me. I mean that when you see a Woman, you 



ought  besides inferring her breadth  to see her length, and to see what we 



call her height; although the last Dimension is infinitesimal in your country. 



If a Line were mere length without ``height,'' it would cease to occupy Space 



and would become invisible. Surely you must recognize this? 







I. I must indeed confess that I do not in the least understand your Lordship. 



\textit{I}. I must indeed confess that I do not in the least understand your Lordship. 



When we in Flatland see a Line, we see length and brightness. If the 



brightness disappears, the Line is extinguished, and, as you say, ceases to 



occupy Space. But am I to suppose that your Lordship gives the brightness the 



title of a Dimension, and that what we call ``bright'' you call ``high''? 







STRANGER. No, indeed. By ``height'' I mean a Dimension like your length: only, 



\textit{Stranger}. No, indeed. By ``height'' I mean a Dimension like your length: only, 



with you, ``height'' is not so easily perceptible, being extremely small. 







I. My Lord, your assertion is easily put to the test. You say I have a Third 



\textit{I}. My Lord, your assertion is easily put to the test. You say I have a Third 



Dimension, which you call ``height.'' Now, Dimension implies direction and 



measurement. Do but measure my ``height,'' or merely indicate to me the 



direction in which my ``height'' extends, and I will become your convert. 



Otherwise, your Lordship's own understanding must hold me excused. 







STRANGER. (To himself.) I can do neither. How shall I convince him? Surely a 



\textit{Stranger}. (To himself.) I can do neither. How shall I convince him? Surely a 



plain statement of facts followed by ocular demonstration ought to suffice.  



Now, Sir; listen to me. 






@ 2443,33 +2443,33 @@ After a long pause he muttered to himself, ``One resource alone remains, if I 



am not to resort to action. I must try the method of Analogy.'' Then followed a 



still longer silence, after which he continued our dialogue. 







SPHERE. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves a 



\textit{Sphere}. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves a 



luminous wake, what name would you give to the wake? 







I. A straight Line. 



\textit{I}. A straight Line. 







SPHERE. And a straight Line has how many extremities? 



\textit{Sphere}. And a straight Line has how many extremities? 







I. Two. 



\textit{I}. Two. 







SPHERE. Now conceive the Northward straight Line moving parallel to itself, 



\textit{Sphere}. Now conceive the Northward straight Line moving parallel to itself, 



East and West, so that every point in it leaves behind it the wake of a 



straight Line. What name will you give to the Figure thereby formed? We will 



suppose that it moves through a distance equal to the original straight line. 



 What name, I say? 







I. A square. 



\textit{I}. A square. 







SPHERE. And how many sides has a Square? How many angles? 



\textit{Sphere}. And how many sides has a Square? How many angles? 







I. Four sides and four angles. 



\textit{I}. Four sides and four angles. 







SPHERE. Now stretch your imagination a little, and conceive a Square in 



\textit{Sphere}. Now stretch your imagination a little, and conceive a Square in 



Flatland, moving parallel to itself upward. 







I. What? Northward? 



\textit{I}. What? Northward? 







SPHERE. No, not Northward; upward; out of Flatland altogether. 



\textit{Sphere}. No, not Northward; upward; out of Flatland altogether. 







If it moved Northward, the Southern points in the Square would have to move 



through the positions previously occupied by the Northern points. But that is 


@ 2490,7 +2490,7 @@ anywhere, so that I could get rid of him  I replied:  



motion which you are pleased to denote by the word `upward'? I presume it is 



describable in the language of Flatland.'' 







SPHERE. Oh, certainly. It is all plain and simple, and in strict accordance 



\textit{Sphere}. Oh, certainly. It is all plain and simple, and in strict accordance 



with Analogy  only, by the way, you must not speak of the result as being a 



Figure, but as a Solid. But I will describe it to you. Or rather not I, but 



Analogy. 


@ 2505,35 +2505,35 @@ One Line produces a Square with four terminal Points. 



Now you can give yourself the answer to your own question: 1, 2, 4, are 



evidently in Geometrical Progression. What is the next number? 







I. Eight. 



\textit{I}. Eight. 







SPHERE. Exactly. The one Square produces a 



\textit{Sphere}. Exactly. The one Square produces a 



SomethingwhichyoudonotasyetknowanameforbutwhichwecallaCube 



with eight terminal Points. Now are you convinced? 







I. And has this Creature sides, as well as Angles or what you call ``terminal 



\textit{I}. And has this Creature sides, as well as Angles or what you call ``terminal 



Points''? 







SPHERE. Of course; and all according to Analogy. But, by the way, not what you 



\textit{Sphere}. Of course; and all according to Analogy. But, by the way, not what you 



call sides, but what we call sides. You would call them solids. 







I. And how many solids or sides will appertain to this Being whom I am to 



\textit{I}. And how many solids or sides will appertain to this Being whom I am to 



generate by the motion of my inside in an ``upward'' direction, and whom you 



call a Cube? 







SPHERE. How can you ask? And you a mathematician! The side of anything is 



\textit{Sphere}. How can you ask? And you a mathematician! The side of anything is 



always, if I may so say, one Dimension behind the thing. Consequently, as 



there is no Dimension behind a Point, a Point has 0 sides; a Line, if I may so 



say, has 2 sides (for the points of a Line may be called by courtesy, its 



sides); a Square has 4 sides; 0, 2, 4; what Progression do you call that? 







I. Arithmetical. 



\textit{I}. Arithmetical. 







SPHERE. And what is the next number? 



\textit{Sphere}. And what is the next number? 







I. Six. 



\textit{I}. Six. 







SPHERE. Exactly. Then you see you have answered your own question. The Cube 



\textit{Sphere}. Exactly. Then you see you have answered your own question. The Cube 



which you will generate will be bounded by six sides, that is to say, six of 



your insides. You see it all now, eh? 






@ 2552,7 +2552,7 @@ contact; not edging to the right nor to the left, but moving somehow out of 



the world, and vanishing into nothing. Soon there was a blank. But still I 



heard the Intruder's voice. 







SPHERE. Why will you refuse to listen to reason? I had hoped to find in you  



\textit{Sphere}. Why will you refuse to listen to reason? I had hoped to find in you  



as being a man of sense and an accomplished mathematician  a fit apostle for 



the Gospel of the Three Dimensions, which I am allowed to preach once only in 



a thousand years: but now I know not how to convince you. Stay, I have it. 


