Euclid's puzzling parallel postulate - Jeff Dekofsky

歐幾里得謎樣的平行公設 - Jeff Dekofsky

Translator: Andrea McDonough Reviewer: Bedirhan Cinar

譯者: Jephian Lin 審譯者: Hao-Wei Chang

就如同各時代 幾何學學生所熟知

As any current or past geometry student knows,

the father of geometry was Euclid,

幾何學之父是歐幾里得。

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

他是一位希臘數學家。 生活在西元前約三百年

埃及的亞歷山卓。

歐幾里得是《幾何原本》

Euclid is known as the author of a singularly influential work

這部影響深遠著作的作者。

known as "Elements."

你覺得你的數學教科書太厚嗎？

You think your math book is long?

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

歐幾里得的《幾何原本》有 13 冊， 並且都只談論幾何。

In "Elements," Euclid structured and supplemented the work

在《幾何原本》中， 歐幾里得建構並補足許多

先前數學家的工作，

of many mathematicians that came before him,

像是畢達哥拉斯（Pythagoras）、

such as Pythagoras, Eudoxus, Hippocrates and others.

歐多克索斯（Eudoxus）、

希波克拉底（Hippocrates）、

還有其它人。

Euclid laid it all out as a logical system of proof

歐幾里得把他們的結果 用邏輯的證明系統寫下來，

這證明系統建構在

built up from a set of definitions,

定義、共同認知、

common notions, and his five famous postulates.

以及他那五個有名的公設之上。 （譯註：公設即一開始就認定要是對的敍述。）

Four of these postulates are very simple and straightforward,

公設中其中四個 是簡單且直覺的，

比如說兩點可以決定一條線。

two points determine a line, for example.

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

而那第五個公設， 則衍伸出我們要講的故事。

這第五個神秘的公設

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

簡單地被稱作「平行公設」。

你看，和前面四個公設不一樣，

You see, unlike the first four,

the fifth postulate is worded in a very convoluted way.

這第五個公設敘述地 十分拐彎抹角。

Euclid's version states that,

這是歐幾里得的版本：

"If a line falls on two other lines

「如果一條直線與另兩條相交

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

並且同一側的的兩內角

加起來小於兩個直角的角度，

add up to less than two right angles,

那麼那兩條直線

終會在那一側相交，

then the lines eventually intersect on that side,

and therefore are not parallel."

因此它們並不平行。」

Wow, that is a mouthful!

哇，這真繞口！

Here's the simpler, more familiar version:

這是比較簡單、大家也比較熟知的版本：

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

「在一平面中，給定一直線及線外的一點，

只能畫出一條新的直線

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

通過這點並和原本的線平行。」

Many mathematicians over the centuries tried to prove the parallel postulate

歷經好幾世紀，許多數學家

試著要用其它四個公設 來證明平行公設，

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

但都失敗了。

在這過程中，他們開始關注於

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

如果第五公設實際上是錯的

邏輯上會有什麼問題。

if the fifth postulate were actually not true.

一些數學史上

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

最偉大的先驅 都考慮過這個問題，

像是海什木（Ibn al-Haytham）、

people like Ibn al-Haytham, Omar Khayyam,

歐瑪爾．海亞姆（Omar Khayyam）、

Nasir al-Din al-Tusi, Giovanni Saccheri,

納速拉丁．圖西（Nasir al-Din al-Tusi）、

幾凡尼．歇克瑞（Giovanni Saccheri）、

鮑耶．亞諾什（Janos Bolyai）、

János Bolyai, Carl Gauss, and Nikolai Lobachevsky.

卡爾．高斯（Carl Gauss）、

尼古拉．羅巴切夫斯基（Nikolai Lobachevsky）。

They all experimented with negating the parallel postulate,

他們都曾試驗過平行公設 錯誤時的情形，

但發現這樣新的假設

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

只會建構出完全不一樣的幾何學。

這些幾何學則合稱為

These geometries became collectively known as non-Euclidean geometries.

「非歐幾何」。

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

嗯，我們會把非歐幾何

的細節留到下一堂課，

但主要的不同在於我們所討論的直線

The main difference depends on the curvature of the surface

所在的曲面 它曲率的不同。

upon which the lines are constructed.

原來歐幾里得 並沒在《幾何原本》中

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

告訴我們完整的故事；

他只是提供了一個可能的方法

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

來看待宇宙。

這取決於我們怎麼看它。

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

平坦的表面是一種樣子，

Flat surfaces behave one way,

而凹的或凸的表面

while positively and negatively curved surfaces

卻表現出很不一樣的特徵。

display very different characteristics.

一開始這些非歐幾何 似乎有點奇怪

At first these alternative geometries seemed strange,

但很快地被發現 它也適切地

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

描述我們的宇宙。

Navigating our planet requires elliptical geometry

在我們的星球上航行 須要橢圓幾何，

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

而同時埃舍爾 （譯註：M. C. Escher 為荷蘭版畫家。）

又以他的藝術 展現雙曲幾何的美。

Albert Einstein used non-Euclidean geometry as well

愛因斯坦也使用非歐幾何

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

在廣義相對論中

來描述時間與空間

在各狀態下如何改變。

as part of his general theory of relativity.

The big mystery is whether Euclid had any inkling

而最大的謎團在於 歐幾里得

在寫下神秘的平行公設時

of the existence of these different geometries

是否注意到這些 不同的幾何學的存在。

when he wrote his postulate.

We may never know,

我們可能永遠不會知道答案，

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

但很難相信

像他這麼聰明的數學家

並對幾何學了解如此透徹，

being the great intellect that he was

他會完全沒有注意到這件事。

and understanding the field as thoroughly as he did.

也許他確實知道，

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

然後故意寫下這樣的平行公設，

好讓好奇的後輩們

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

來發現其細節內容。

If so, he's probably pleased.

如果是這樣，他也許對結果很滿意。

These discoveries could never have been made

若沒有那些天才又求新求變的思想家，

without gifted, progressive thinkers

他們可以屏棄一些先入為主的觀點

able to suspend their preconceived notions

並獨立思考，

這些理論可能永遠不會被發現。

and think outside of what they've been taught.

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

我們也應該樂於

偶爾放下既有的概念 或是物理上的經驗

and physical experiences

來看看更廣的世界，

and look at the larger picture,

or we risk not seeing the rest of the story.

否則我們將錯過許多事情。