Euclid's puzzling parallel postulate - Jeff Dekofsky

Translator: Andrea McDonough Reviewer: Bedirhan Cinar

翻译人员: Minji Seo 校对人员: Yolanda Zhang

不论是哪个时代的几何学学生都知道，

As any current or past geometry student knows,

the father of geometry was Euclid,

几何学之父是欧几里得，

a Greek mathematician who lived in Alexandria, Egypt, around 300 B.C.E.

他是一位希腊数学家， 生活在西元前约三百年的

埃及亚历山大。

欧几里得撰写了

Euclid is known as the author of a singularly influential work

《几何原本》这部影响深远的著作。

known as "Elements."

你觉得你的数学教材太厚吗？

You think your math book is long?

Euclid's "Elements" is 13 volumes full of just geometry.

欧几里得的《几何原本》有 13 册， 并且都只谈论几何。

In "Elements," Euclid structured and supplemented the work

在《几何原本》中， 欧几里得建构并

补足了许多先前数学家的工作，

of many mathematicians that came before him,

像是毕达哥拉斯 （Pythagoras）、

such as Pythagoras, Eudoxus, Hippocrates and others.

欧多克索斯 （Eudoxus）、

希波克拉底 （Hippocrates）、

等等。

Euclid laid it all out as a logical system of proof

欧几里得把他们的结果 通过被证明过的逻辑体系进行编撰，

这一体系建立在定义、

built up from a set of definitions,

共同认知、

common notions, and his five famous postulates.

以及他那五个有名的公理之上 （译注：即一开始被认定为正确的叙述。）

Four of these postulates are very simple and straightforward,

其中四个公理非常简单易懂，

比如说两点可以决定一条线。

two points determine a line, for example.

The fifth one, however, is the seed that grows our story.

而那第五个公理， 则衍生出我们要讲的故事。

这第五个神秘的公理

This fifth mysterious postulate is known simply as the parallel postulate.

被简单地称作“平行公理”。

可以看到，和前面四个公理不一样，

You see, unlike the first four,

the fifth postulate is worded in a very convoluted way.

第五个公理的描述十分拐弯抹角。

Euclid's version states that,

欧几里得的版本是：

"If a line falls on two other lines

“如果一条直线与另两条直线相交

so that the measure of the two interior angles on the same side of the transversal

并且同一侧的两内角

加起来小于两个直角的角度，

add up to less than two right angles,

那么这两条直线

最终会在那一侧相交，

then the lines eventually intersect on that side,

and therefore are not parallel."

因此它们并不平行。”

Wow, that is a mouthful!

哇，这真绕口！

Here's the simpler, more familiar version:

而这是比较简单、众所周知的版本：

"In a plane, through any point not on a given line,

“在一个平面中， 给定一直线及线外的一点，

只能画出一条通过

only one new line can be drawn that's parallel to the original one."

这点并与已知直线平行的线。”

Many mathematicians over the centuries tried to prove the parallel postulate

历经好几世纪，许多数学家

试着要用其它四个公理来证明平行公理，

from the other four, but weren't able to do so.

但都失败了。

在这个过程中，他们不禁想到

In the process, they began looking at what would happen logically

如果第五个公理实际上是错的，

逻辑上会有什么问题。

if the fifth postulate were actually not true.

一些数学史上

Some of the greatest minds in the history of mathematics ask this question,

最伟大的先驱都考虑过这个问题，

像是海什木 （Ibn al-Haytham）、

people like Ibn al-Haytham, Omar Khayyam,

欧玛尔．海亚姆 （Omar Khayyam）、

Nasir al-Din al-Tusi, Giovanni Saccheri,

纳速拉丁．图西 （Nasir al-Din al-Tusi）、

几凡尼．歇克瑞 （Giovanni Saccheri）、

鲍耶．亚诺什 （Janos Bolyai）、

János Bolyai, Carl Gauss, and Nikolai Lobachevsky.

卡尔．高斯 （Carl Gauss）、

以及尼古拉．罗巴切夫斯基 （Nikolai Lobachevsky）。

They all experimented with negating the parallel postulate,

他们都试验过平行公理错误时的情形，

但发现这种猜测

only to discover that this gave rise to entire alternative geometries.

只会建构出完全不一样的几何学。

这些几何学则合称为

These geometries became collectively known as non-Euclidean geometries.

“非欧几何”。

We'll leave the details of these different geometries for another lesson.

不过我们会把

非欧几何的细节留到下一堂课，

但主要的不同在于我们所讨论的

The main difference depends on the curvature of the surface

直线所在曲面的曲率不同。

upon which the lines are constructed.

原来，欧几里得并没在《几何原本》中

Turns out Euclid did not tell us the entire story in "Elements,"

告诉我们完整的故事；

他只是提供了一个可能的方法

and merely described one possible way to look at the universe.

来看待宇宙。

这取决于我们看待它的视角。

It all depends on the context of what you're looking at.

平坦的表面是一种样子，

Flat surfaces behave one way,

而凹的或凸的表面

while positively and negatively curved surfaces

却表现出很不一样的特征。

display very different characteristics.

这些非欧几何一开始似乎有点奇怪，

At first these alternative geometries seemed strange,

但很快地被发现它也能恰当地

but were soon found to be equally adept at describing the world around us.

描述我们的宇宙。

Navigating our planet requires elliptical geometry

在我们的星球上航行需要用到椭圆几何，

while the much of the art of M.C. Escher displays hyperbolic geometry.

而同时埃舍尔 （译注：荷兰版画家）

又以他的艺术展现了双曲几何的美。

Albert Einstein used non-Euclidean geometry as well

爱因斯坦也使用非欧几何

to describe how space-time becomes warped in the presence of matter,

在广义相对论中

来描述时间与空间

在各状态下如何改变。

as part of his general theory of relativity.

The big mystery is whether Euclid had any inkling

而最大的谜团在于欧几里得

在写下神秘的平行公理时，

of the existence of these different geometries

是否注意到这些不同几何学的存在。

when he wrote his postulate.

We may never know,

我们可能永远不会知道答案，

but it's hard to believe he had no idea whatsoever of their nature,

但很难相信

像他这么聪明的数学家

对几何学又了解得如此透彻，

being the great intellect that he was

怎么会完全没有注意到这件事。

and understanding the field as thoroughly as he did.

也许他确实知道，

Maybe he did know and he wrote the postulate in such a way

然后故意写下这样的平行公理，

好让好奇的后辈们

as to leave curious minds after him to flush out the details.

来发现背后的细节。

If so, he's probably pleased.

如果是这样，他也许会感到很欣慰。

These discoveries could never have been made

若没有那些不断创新的

without gifted, progressive thinkers

天才思想家能够摒弃一些

able to suspend their preconceived notions

先入为主的观点并独立思考，

这些理论可能永远不会被发现。

and think outside of what they've been taught.

We, too, must be willing at times to put aside our preconceived notions

我们也应该乐于

偶尔放下既有的概念和物理经验，

and physical experiences

来看看更广的世界，

and look at the larger picture,

or we risk not seeing the rest of the story.

否则我们可能会错过许多奇妙的事情。