the father of geometry was Euclid,

幾何學之父是歐幾里得。

a Greek mathematician who lived in Alexandria, Egypt

他是一位希臘數學家。 生活在西元前約三百年

around 300 B.C.E.

埃及的亞歷山卓。

Euclid is known as the author

歐幾里得是《幾何原本》

of a singularly influential work known as Elements.

這部影響深遠著作的作者。

You think your math book is long?

你覺得你的數學教科書太厚嗎？

Euclid's Elements is 13 volumes filled 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

在《幾何原本》中， 歐幾里得建構並補足許多

the work of many mathematicians that came before him,

先前數學家的工作，

such as Pythagoras,

像是畢達哥拉斯（Pythagoras）、

Eudoxus,

歐多克索斯（Eudoxus）、

Hippocrates,

希波克拉底（Hippocrates）、

and others.

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，

built up from a set of definitions,

這證明系統建構在

common notions,

定義、共同認知、

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，

two points determine a line, for example.

比如說兩點可以決定一條線。

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。

This fifth mysterious postulate is known

這第五個神秘的公設

simply as the "Parallel Postulate".

簡單地被稱作「平行公設」。

You see, unlike the first four,

你看，和前面四個公設不一樣，

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

並且同一側的的兩內角

on the same side of the transversal

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

then the lines eventually intersect on that side,

終會在那一側相交，

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，

only one new line can be drawn

只能畫出一條新的直線

that's parallel to the original one."

通過這點並和原本的線平行。」

Many mathematicians over the centuries

歷經好幾世紀，許多數學家

tried to prove the parallel postulate from the other four,

試著要用其它四個公設 來證明平行公設，

but weren't able to do so.

但都失敗了。

In the process, they began looking

在這過程中，他們開始關注於

at what would happen logically

如果第五公設實際上是錯的

if the fifth postulate were actually not true.

邏輯上會有什麼問題。

Some of the greatest minds

一些數學史上

in the history of mathematics ask this question,

最偉大的先驅 都考慮過這個問題，

people like Ibn al-Haytham,

像是海什木（Ibn al-Haytham）、

Omar Khayyam,

歐瑪爾．海亞姆（Omar Khayyam）、

Nasir al-Din al-Tusi,

納速拉丁．圖西（Nasir al-Din al-Tusi）、

Giovanni Saccheri,

幾凡尼．歇克瑞（Giovanni Saccheri）、

Janos Bolyai,

鮑耶．亞諾什（Janos Bolyai）、

Carl Gauss,

卡爾．高斯（Carl Gauss）、

and Nikolai Lobachevsky.

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，

only to discover that this gave rise

但發現這樣新的假設

to entire alternative geometries.

只會建構出完全不一樣的幾何學。

These geometries became collectively known

這些幾何學則合稱為

as Non-Euclidean Geometries.

「非歐幾何」。

Well, we'll leave the details

嗯，我們會把非歐幾何

of these different geometries for another lesson,

的細節留到下一堂課，

the main difference depends on the curvature

但主要的不同在於我們所討論的直線

of the surface upon which the lines are constructed.

所在的曲面 它曲率的不同。

Turns out that Euclid did not tell us

原來歐幾里得 並沒在《幾何原本》中

the entire story in Elements;

告訴我們完整的故事；

he merely described one possible way

他只是提供了一個可能的方法

to look at the universe.

來看待宇宙。

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

這取決於我們怎麼看它。

Flat surfaces behave one way,

平坦的表面是一種樣子，

while positively and negatively curved surfaces

而凹的或凸的表面

display very different characteristics.

卻表現出很不一樣的特徵。

At first these alternative geometries seemed a bit strange

一開始這些非歐幾何 似乎有點奇怪

but were soon found to be equally adept

但很快地被發現 它也適切地

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

而同時埃舍爾 （譯註：M. C. Escher 為荷蘭版畫家。）

displays hyperbolic geometry.

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 the way that space time

在廣義相對論中

becomes work in the presence of matter

來描述時間與空間

as part of his General Theory of Relativity.

在各狀態下如何改變。

The big mystery here is whether or not Euclid

而最大的謎團在於 歐幾里得

had any inkling of the existence of these different geometries

在寫下神秘的平行公設時

when he wrote his mysterious postulate.

是否注意到這些 不同的幾何學的存在。

We may never know the answer to this question,

我們可能永遠不會知道答案，

but it seem hard to believe

但很難相信

that he had no idea whatsoever of their nature,

像他這麼聰明的數學家

being the great intellect that he was

並對幾何學了解如此透徹，

and understanding the field as thoroughly as he did.

他會完全沒有注意到這件事。

Maybe he did know

也許他確實知道，

and intentionally wrote the Parallel Postulate in such a way

然後故意寫下這樣的平行公設，

as to leave curious minds after him

好讓好奇的後輩們

to flush out the details.

來發現其細節內容。

If so, he's probably quite 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

who are able to suspend their preconceived notions

並獨立思考，

and think outside of what they have been taught.

這些理論可能永遠不會被發現。

We, too, must be willing at times

我們也應該樂於

to put aside our preconceived notions and physical experiences

偶爾放下既有的概念 或是物理上的經驗

and look at the larger picture,

來看看更廣的世界，

or we risk not seeing the rest of the story.

否則我們將錯過許多事情。