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 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? 

Kevin：你打量、触碰够了么？你还没介绍你自己呢？

 Rick：最、最、最杰出的先生，请原谅我的笨拙吧，我不小心触碰你，并不是对法律的无知，而是因为我对您这次特殊的拜访而感到惊讶和紧张。我也求求你了，不要把这件事泄露给任何人，特别是我的妻子！但求求你了，神父，在我们进一步探讨之前，能不能满足您小百姓的好奇心，告诉他您是从哪里来的呢？

Stranger. From Space, from Space, Sir: whence else? I. Pardon me, my Lord, but is not your Lordship already in Space, your Lordship and his humble servant, even at this moment?

Kevin：我来自空间，空间，先生。还能来自哪呢？

Rick：但是，神父，您现在不已经在空间里了吗，您和您谦卑的百姓，现在不是就在这里吗？

 Stranger. Pooh! what do you know of Space? Define Space. I. Space, my Lord, is height and breadth indefinitely prolonged.

Kevin：哈，你知道什么是空间？说说空间吧。

Rick：我的神父，空间是一个长度和宽度可以无限延长的地方。

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

Kevin：看吧，你都不知道什么是空间。你以为它只是个二维的而已，但我就来告诉你三个维度：高，宽，长。

Rick：我的神父，您一定很爱开玩笑。我们也可以把长度和宽度，用另外一个词讲出来，也就是说可以有四个名字表达，但其实还是长和宽而已。

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 Third Dimension, unknown to me?

Kevin：但我并不是说三个名字，而是三个维度。

Rick：那尊敬的神父，您能给我讲一讲那个方向，也就是第三维吗？

Stranger. I came from it. It is up above and down below. I. My Lord means seemingly that it is Northward and Southward.

Kevin：我来自三维空间，第三维就是指向上和向下。

Rick：我的神父，您的意思应该是向北和向南。

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

Kevin：并不。我说的是另外一个你看不到的方向，因为你没有在里头的眼睛。 Rick：不好意思，我的神父，您只要观察一会会，就能发现我有一个在两端很完美的发光体……

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.

Kevin：是的。但为了能看到空间，你需要有一个眼睛，并不是在两端的眼睛，而是需要在你边上，也就是你的里头，用我们立体世界的称呼叫你的边上。 Rick：什么，一个长在里头的眼睛，一个在胃里的眼睛！神父真会说笑。

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 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.

Kevin：我没有开玩笑。我说我来自立体世界，或者既然你不能理解什么叫立体世界，我们可以称呼为三维空间。从我这看下来，你们只在一个平面，虽然你们称之为空间。以我的角度来看，你们称为‘立体’的从四边封闭的东西，比如你们的房子，教堂，金库，保险箱，甚至你身体里头比如胃，全都暴露在我眼前。

Rick：这种断言也就随便说说啦，我的神父。

 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 remain a while with you and then retire to his room, leaving you and your Wife 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.

Kevin：你是说不容易被证明？但我就是来证明的！当我来你这，我看到你的四个儿子，都是五边形，都在自家公寓里头，我也看到你的两个六边形孙子；最小的那个孙子和你们呆了一会然后去他自己房间了，只留下你和你妻子。我还看见三个等腰三角形仆人在厨房准备晚餐，还有一个男仆人在水槽。然后我来了，你想想我怎么来的？

Rick：我想是从屋顶来的吧

Strange. 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 humble servant might be easily ascertained by any one of the neighbourhood possessing your Lordship's ample means of information.

Kevin：并不是，你的屋顶，你应该清楚，是刚刚修复好的，没有任何的漏洞，甚至连尖锐的女人都没法穿过。我告诉过你，我来自空间。在我这么清晰的描述了你的孩子和家居日常这些情况后，你信服了吧？

Rick：我的神父，您一定清楚，这些关于我的信息，很简单就可以从我的邻居那里得到。

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 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 Straight Line, is, really and scientifically, a very thin Parallelogram, possessing Two Dimensions, like the rest of us, viz., length and breadth (or thickness).

Kevin：（自言自语）我还怎么做呢？等等，我想到一个新的论点。（对着Rick）当你看到一条直线，比如你的妻子，你觉得是几个维度？ Rick：你一定认为我是一个在数学方面无知的人，其实我不是，女人应该是一个直线，只有一维？不不不，我的神父，我们正方形跟你们原先一样是很发达的，我们也知道女人，我们叫她为直线，但从真实的，科学的讲，是一个很细很细的平行四边形，也就是二维的，就像我们一样，也就是长和宽。

Stranger. But the very fact that a Line is visible implies that it possesses yet another Dimension. I. My Lord, I have just acknowledged 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.

Kevin：但事实上一条线虽然只有一个维度可见，却暗示了其他维度的存在。 Rick：我的神父，我刚刚承认了，女人既又长又宽。我们看得见她的长，也可以推论出它的宽；虽然特别的少，但也可以被量出。

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. 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"?

Kevin：你没明白我的意思。当你看到一个女士的长度，你除了推断出她的宽度，还应该可以‘看出’她的高度，虽然从你们的角度看，这个‘高’无穷小。如果一条线仅仅只有长度，那么它就不会占据任何空间，也就是不可见的。你能理解这个么？

Rick：我必须坦白，我根本没有听懂你在说什么，我的神父。当我们看见一个在平面国的直线，我们可以看见它的长度和亮度，如果亮度消失了，这条线就看不见了，也就是说停止占据空间了。那我可以这样认为吗，也就是您把亮度变成了一个新的维度，所以也就是你说的“高度”，就是“亮度”？

 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 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 understand must hold me excused.

Kevin：不，不。事实上，我说的‘高度’和你们说的长度是一个类似的维度。只是你们察觉不到‘高’，因为太小了。

Rick：我的神父，我们很容易测试您的主张正确与否。你说我有第三个维度，就是你所谓的‘高’。但，维度意味着方向和度量。如果你能量出我的高度，或者告诉我我的‘高’往哪个方向延伸，我就相信你的理论。否则，我没法相信你。

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. You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, or in, the top of which you and your countrymen move about, without rising above or falling below it.

Kevin：（自言自语）两种我都可以做。但我怎么才能说服他呢？一个可见的示范加上一个浅显的结论应该够了吧。（转向Rick）听着，先生。 你们生活在一个平面上，姑且称为平面国，是一个巨大的平面，你们国民在这个平面上移动，却没有升高于平面或者掉落低于平面。

I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from aPoint to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I make in your plane aSECTION which you, very rightly, call a Circle. For even a Sphere--which is my proper name in my own country--if he manifest himself at all to an inhabitant of Flatland-- must needs manifest himself as a Circle.

Kevin：我不是一个平面图形，而是一个立体的。你以为我是圆形，但事实上我并不是一个圆形，而是由无数个圆形组成，从仅仅是一个点的圆到直径13英寸的园，一个叠一个组成。当我和你们所在的平面相交，正如我现在做的这样，我在你们平面国就展示了一个截面，正如你所称呼，是一个圆形。比如当一个球体，我在我们立体世界就是一个球体，要在平面国显现的时候，就只能是显示成为一个圆形。

Do you not remember--for I, who see all things, discerned last night the phantasmal vision of Lineland written upon your brain-- do you not remember, I say, how when you entered the realm of Lineland, you were compelled to manifest yourself to the King, not as a Square, but as a Line, because that Linear Realm had not Dimensions enough to represent the whole of you, but only a slice or SECTION of you? In precisely the same way, your country of Two Dimensions is not spacious enough to represent me, a being of Three, but can only exhibit a slice or SECTION of me, which is what you call a Circle. Kevin：你不记得了么？我能看到所有的事情，比如你昨晚脑子里像幽灵一样浮现的“线性世界”。你不记得么，我是说当你进入线性世界的时候，你显现在他们国王面前的不是一个正方形而是一条线，因为线性世界的维度不足以显示整个你，而只能是你的一个切片或者说截面。同样的，你们这个世界的两维也不足以完整的展示我这个三维立体，但可以显示我的一个切片或者说截面，也就是你看到的圆形。

The diminished brightness of your eye indicates incredulity. But now prepare to receive proof positive of the truth of my assertions. You cannot indeed see more than one of my SECTIONs, or Circles, at a time; for you have no power to raise your eye out of the plane of Flatland; but you can at least see that, as I rise in Space, so my SECTIONs become smaller. See now, I will rise; and the effect upon your eye will be that my Circle will become smaller and smaller till it dwindles to a point and finally vanishes.

Kevin：你暗淡的眼神说明你还不相信，但请准备好迎接能证明我论断的更多正面例证。因为你没法从你们平面国的平面上升起，所有你没法同时看到我的多个截面，也就是多个圆；但你会发现当我升起的时候，我的截面会变小。现在，请看，我升起来一些，我在你们平面国呈现的圆越来越小，最后变成一个点，直到消失。

There was no "rising" that I could see; but he diminished and finally vanished. I winked once or twice to make sure that I was not dreaming. But it was no dream. For from the depths of nowhere came forth a hollow voice--close to my heart it seemed—

"Am I quite gone? Are you convinced now? Well, now I will gradually return to Flatland and you shall see mySECTION become larger and larger."

Kevin：我是不是消失了？现在你相信了吧？好的，现在呢，我再慢慢回到你们平面国，你就会看到我的截面越来越大。

Every reader in Spaceland will easily understand that my mysterious Guest was speaking the language of truth and even of simplicity. But to me, proficient though I was in Flatland Mathematics, it was by no means a simple matter. The rough diagram given above will make it clear to any Spaceland child that the Sphere, ascending in the three positions indicatedthere, must needs have manifested himself to me, or to any Flatlander, as a Circle, at first of full size, then small, and at last very small indeed, approaching to a Point. But to me, although I saw the facts before me, the causes were as dark as ever. All that I could comprehend was, that the Circle had made himself smaller and vanished, and that he had now reappeared and was rapidly making himself larger. When he regained his original size, he heaved a deep sigh; for he perceived by my silence that I had altogether failed to comprehend him. And indeed I was now inclining to the belief that he must be no Circle at all, but some extremely clever juggler; or else that the old wives' tales were true, and that after all there were such people as Enchanters and Magicians. 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.

Kevin：还有一个论据，要是我不诉诸于实践证明，我们试着来做个理论推论。 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.

Kevin：告诉我，数学家先生，如果一个点向北移动，并留下带着发亮的尾迹。你怎么称呼这一个尾迹？

Rick：一条直线。

Sphere. And a straight Line has how many extremities? I. Two.

Kevin：那么一条直线有几个端点？ Rick：两个。

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.

Kevin：好，我们现在想象一下这个北向的直线沿着东西方向移动，直线上的每一个点都在其身后留下尾迹也就是直线。那么你们将这个图像叫做什么？我们假定直线移动的距离等于它自身，什么名字？

Rick： 一个正方形。

Sphere. And how many sides has a Square? How many angles? I. Four sides and four angles.

Kevin：那么，正方形有几个边，几个角？

 Rick：四条边和四个角。

Sphere. Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself upward. I. What? Northward?

Kevin：那么再进一步想象一下，平面国的一个正方形，平行于它自身的方向移动。

 Rick：什么方向移动？向北边么？

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 not my meaning. I mean that every Point in you--for you are a Square and will serve the purpose of my illustration--every Point in you, that is to say in what you call your inside, is to pass upwards through Space in such a way that no Point shall pass through the position previously occupied by any other Point; but each Point shall describe a straight Line of its own. This is all in accordance with Analogy; surely it must be clear to you. Restraining my impatience--for I was now under a strong temptation to rush blindly at my Visitor and to precipitate him into Space, or out of Flatland, anywhere, so that I could get rid of him--I replied:-"And what may be the nature of the Figure which I am to shape out by this motion which you are pleased to denote by the word `upward'? I presume it is describable in the language of Flatland."

Kevin：不，不是向北，而是向上，完全远离平面国。如果只是向北移动，那么四边形南部的点会经过之前四边形北部的点所经过的地方，但这不是我所讲的方向。我的意思是你的每一个点 ---刚好你是个四边形，可以代表我的模拟 --- 你的每一个点，正如你们所讲的你的内部，用一种方式向上移动，而且每个点都不会经过其他点之前经过的地方，而每个点都会描绘出一条直线出来。这么推论下来，你应该就清楚了吧。

Rick：那么我这个四边形用你说的‘向上’移动划出的图案叫什么呢？我是说用我们平面国的语言描述。

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. We began with a single Point, which of course--being itself a Point-has only ONE terminal Point. One Point produces a Line with TWO terminal Points. 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.

Kevin：哦，当然，这很简单。单从推论而言，你不能说四边形划出的结果是个‘图案’，而是一个立体。我来描述给你听，或者也不是我来描述，而是推论而言。我们先从一个点开始，当然点自身就只有一个端点。一个点生成一条带着两个端点的线。一条线生成带着四个端点的正方形。那么你可以试着自己回答一下你的问题：1，2，4…显然是一个几何级数。那么下一个数是？

Rick：八。

Sphere. Exactly. The one Square produces a SOMETHING-WHICHYOU- DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WECALL-A-CUBE with EIGHT terminal Points. Now are you convinced?

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

Kevin：完全正确。一个正方形生成了一个你还不知道名字的一个东东，我们姑且称之为‘Cube立方体’，它包含了八个点。现在你信服了么？

Rick：那这个奇怪的边，或是这个奇怪的角，有末端的点吗？

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 generate by the motion of my inside in an "upward" direction, and whom you call a Cube?

Kevin：当然，这都要靠类比，类推。但是，并不是你们所称之为的侧面，就是我们所称 之为侧面，我们的侧面，你们应该会叫它为立体。

Rick:如果这种把我的本体往上移动的这个过程发生在我的身上的话，那我会有几条边， 几个面呢? Rick：如果这种把我的本体往上移动的这个过程发生在我的身上的话，那我会有几条边，几个面呢？

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.

Kevin：你怎么能问我呢?你可是一个数学家啊!任何事物的侧面，如果我可以这样说的 话，背后的逻辑都是一个维度的。总而言之，一个点是没有纬度的，所以一个点有零个侧 面;一条线，如果可以这样说的话，有两个侧面;一个正方形有四条边;零，二，四;这 种方法你们叫做什么?

Rick：算术的。

Sphere. And what is the next number? I. Six.

 Kevin： 那么下一个数字呢？

Rick：六

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? "Monster," I shrieked, "be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries. Either thou or I must perish." And saying these words I precipitated myself upon him.

Kevin：完全正确。那么你看你已经回答了你自己的问题。那么这个产生的立方体就会有六条 边，也就是说六个你的里面，你现在明白了吗，嗯?

Rick：妖怪，你个变戏法的败类，你个巫师，你个噩梦，魔鬼，我已经忍受不了你这种嘲弄了，不是你死就是我亡。