為什麼圓周率差點成了6.283185...：英語精讀講義

I'm sure that you're already familiar with the whole pi versus tau debate.

相信大家對 pi 和 tau 的爭論已經不陌生了。

A lot of people say that the fundamental circle constant

許多人認為，作為基礎的圓常數，

we hold up should be the ratio 片語動詞

我們所推崇的應該是

of a circle's circumference to its radius,

圓的周長與其半徑的比值，

which is around 6.28,

也就是大約 6.28，

not the ratio to its diameter,

而不是周長與直徑的比值，

the more familiar 3.14.

即大家更熟悉的 3.14。

These days, we often call that larger constant tau,

如今，我們常把那個更大的常數稱為 tau，

popularized by Michael Hartl's Tau Manifesto.

這一稱呼由 Michael Hartl 的《Tau 宣言》所普及。

Although personally,

雖然就我個人而言，

I'm quite partial to Robert Palais's proposed notation 片語搭配

我相當偏愛 Robert Palais 提議的符號，

of a pi with three legs.

也就是長了三條腿的 pi。

In either of these manifestos

無論是在這兩份宣言中，

and on many many other places of the internet,

還是在網路上的許多其他地方，

you can read to no end 慣用語

你都可以讀到數不勝數的文章，

about how many formulas look a lot cleaner using tau,

講述使用 tau 能讓多少公式看起來簡潔得多，

largely because the number of radians

很大程度上是因為，

describing a given fraction of a circle

在描述圓中特定比例的弧度數值時，

is actually that fraction of tau.

恰好就是 tau 的該等比例。

That dead horse is beat. 慣用語

這個話題已經被說爛了。

I'm not here to make that case further. 片語搭配

我今天不是來繼續論證這一點的。

Instead,

相反，

I'd like to talk about the seminal moment in history 用詞

我想談談歷史上的那個關鍵時刻：

when pi as we know it became the standard.

也就是我們所熟知的 pi 成為標準常數的時候。

For this,

為此，

one fruitful place to look

一個非常值得探索的方向，

is at the old notes and letters

就是去查閱那些舊筆記和信件，

by one of history's most influential mathematicians,

它們出自歷史上最具影響力的數學家之一：

Leonhard Euler.

李昂哈德·歐拉（Leonhard Euler）。

Luckily, we now have an official

幸運的是，我們現在有一位官方的

3Blue1Brown Switzerland correspondent, Ben Hambrecht,

3Blue1Brown 駐瑞士記者 Ben Hambrecht，

who was able to go to the library

他有幸前往

in Euler's hometown

歐拉家鄉的圖書館，

and get his hands on some of the original documents. 慣用語

查閱到了一些原始文獻。

And in looking through some of those,

在翻閱其中的部分文獻時，

it might surprise you to see Euler write, 表達脈絡

你可能會驚訝地發現歐拉寫道：

"Let pi be the circumference of a circle

「令 pi 為一個圓的周長，

whose radius is one."

其半徑為 1。」

That is the 6.28 constant

也就是說，這個 6.28 常數

that we would now call tau.

就是我們現在所說的 tau。

And it's likely he was using the Greek letter pi 文化背景

而且他當時很可能是將希臘字母 pi

as a P for perimeter.

作為周長（perimeter）單字首字母 P 來使用的。

So was it the case that Euler,

那麼情況是否真的是，

genius of the day,

作為當代天才的歐拉，

was more notationally enlightened than the rest of the world,

在符號學上比世界上其他人更加開明，

fighting the good fight for 6.28? 慣用語

為推廣 6.28 而不懈努力呢？

And if so,

如果真是如此，

who's the villain of our story, 表達脈絡

那麼我們這個故事裡的反派是誰呢？

pushing the 3.1415 constant

是誰在推行 3.1415 這個常數，

shoved in front of most students today?

並將其強塞給如今的大多數學生呢？

Well,

好吧，

the work that really established pi as we now know it 片語搭配

真正將我們現在所知的 pi

as the commonly recognized circle constant

確立為公認圓常數的著作，

was an early calculus book from 1748.

是 1748 年出版的一本早期微積分教材。

At the start of chapter 8,

在第 8 章的開頭，

in describing the semi-circumference of a circle with radius one,

在描述半徑為 1 的圓的半周長時，

and after expanding out a full 128 digits of this number,

並且在將這個數整整展開了 128 位數字之後，

one of them wrong by the way,

（順便一提，其中有一位寫錯了），

the author adds,

作者補充道：

"Which for the sake of brevity, 片語搭配

「為了簡便起見，

I may write pi."

我可以將其寫為 pi。」

Now, there were other texts and letters here and there

誠然，當時各處還有其他的文獻和信件，

with varying conventions for the notation

對於各種與圓有關的常數的符號

of various circle constants,

有著不同的使用習慣，

but this book and this section in particular

但正是這本書，尤其是這一節，

was really the one to spread the notation throughout Europe 文法

真正將這個符號傳播到了整個歐洲，

and eventually the world.

並最終傳遍了全世界。

So,

那麼，

what monster wrote this book 語氣意圖

究竟是哪個「怪物」寫了這本書，

with such an unprincipled take towards circle constants?

對圓常數採取如此毫無原則的態度？

Well,

嗯，

Euler again.

還是歐拉。

In fact,

事實上，

if you look further,

如果你進一步挖掘，

you can find instances of Euler using the symbol pi

你還會發現歐拉使用符號 pi

to represent a quarter turn of the circle,

來表示四分之一圓周的實例，

what we would call today pi halves or tau fourths. 片語搭配

也就是我們今天所說的二分之 pi 或四分之 tau。

In fact, Euler's use of the letter pi

事實上，歐拉對字母 pi 的使用

seems to be much more analogous to our use of the Greek letter theta. 用詞

似乎更類似於我們對希臘字母 theta 的使用。

It's typical for us to let it represent an angle,

我們通常讓它代表一個角度，

but no one angle in particular.

但並不特指某一個特定的角度。

Sometimes it's 30 degrees,

有時它是 30 度，

maybe other times it's 135,

也許其他時候是 135 度，

and most times it's just a variable for a general statement.

大多數時候它只是用於一般性陳述的變數。

It depends on the problem and the context before us.

這取決於我們面前的問題和語境。

Likewise, Euler let pi represent whatever circle constant

同樣地，歐拉讓 pi 代表任何

best suited the problem before him.

最適合他眼前問題的圓常數。

Though it's worth pointing out 表達脈絡

不過值得指出的是，

that he typically framed things in terms of unit circles with radius one, 片語搭配

他通常是以半徑為 1 的單位圓為基礎來建構理論的，

so the 3.1415 constant

所以 3.1415 這個常數

would almost always have been thought of as the ratio 文法

幾乎總是被認為代表了

of a circle's semi-circumference to its radius,

圓的半周長與半徑的比值，

none of this circumference to its diameter nonsense. 語氣意圖

而絕不是周長與直徑比值這種無聊的東西。

And I think Euler's use of this symbol

我認為歐拉對這個符號的使用

carries with it a general lesson 慣用語

給我們帶來了一個普適的啟示，

about how we should approach math.

關於我們應該如何對待數學。

The thing you have to understand about Euler 文法

關於歐拉，你必須了解的一點是，

is that this man solved problems,

這個人一直在解決問題，

a lot of problems.

解決大量的問題。

I mean,

我的意思是，

day in day out, 慣用語

日復一日，

breakfast, lunch, and dinner,

不論是早飯、午飯還是晚飯時間，

he was just churning out puzzles and formulas 片語動詞

他總是在不斷產出各種難題和公式，

and having insights and creating entire new fields left and right. 慣用語

並接連獲得洞見，隨手開創出一個個全新的領域。

Over the course of his life,

在他的一生中，

he wrote over 500 books and papers,

他寫了超過 500 本書和論文，

which amounted to a rate of 800 pages per year. 片語搭配

算下來以每年 800 頁的速度產出。

And these are dense math pages.

而且這些都是資訊密集的數學文獻。

And then,

此外，

after his death,

在他去世後，

another 400 publications surfaced.

又有 400 份出版物浮出水面。

It's often joked that formulas in math 文化背景

人們經常開玩笑說，數學中的公式

have to be named after the second person to prove them,

必須以第二個證明它的人來命名，

because the first is always going to be Euler.

因為第一個證明的人總是歐拉。

His mind was not focused on what circle constant 表達脈絡

他的心思並沒有集中在

we should take as fundamental.

究竟該把哪個圓常數視作最基礎的問題上。

It was on solving the task sitting in front of him

他關注的是解決在那一刻

in a particular moment

正擺在他面前的具體任務，

and writing a letter to the Bernoullis

然後在事後給伯努利家族寫封信

to boast about doing so afterwards.

去吹噓自己搞定了。

For some problems, 表達脈絡

對於某些問題，

the quarter circle constant was most natural to think about.

採用四分之一圓常數是最自然的思考方式。

For others,

對於其他問題，

the full circle constant.

則是全圓常數。

And for others still,

而對於還有些問題，

say at the start of chapter 8 of his famous calculus book,

比如在他著名微積分教材第 8 章的開頭，

maybe the half circle constant

也許半圓常數

was most natural to think about.

才是最自然的思考方式。

Too often in math education,

在數學教育中，往往過於

the focus is on which of multiple competing views 表達脈絡

把焦點放在關於某個主題的多個相互競爭的觀點中

about a topic is right.

究竟哪一個是正確的。

Is it correct to say

我們應該說

that the sum of all positive integers is -1/12?

所有正整數之和為 -1/12 是正確的嗎？

Or is it correct to say

還是應該說

that it diverges to infinity?

它發散到無窮大才是正確的？

Can the infinitesimal values of calculus

微積分中的無窮小量

be taken literally, 片語搭配

可以從字面上來理解嗎，

or is it only correct to speak in terms of limits?

還是說只能用極限的語言來表述才正確？

Are you allowed to divide a number by zero?

你能夠允許把一個數除以零嗎？

These questions in isolation 片語搭配

孤立地看待這些問題，

just don't matter.

其實根本無關緊要。

Our focus should be on specific problems and puzzles,

我們的重點應該放在具體的問題和謎題上，

both those of practical application

既包括那些有實際應用價值的，

and those of idle pondering for knowledge's own sake. 片語搭配

也包括那些純粹為了求知而進行的悠閒思考。

Then,

那麼，

when questions of standards arise,

當標準問題出現時，

you can answer them with respect to a given context. 片語搭配

你就可以根據給定的語境來回答它們了。

And inevitably,

並且不可避免地，

different contexts will lend themselves 片語搭配

不同的語境自然會

to different answers

引導出不同的答案，

of what seems most natural.

以最自然的方式呈現。

But that's okay.

但這沒關係。

Outputting 800 pages a year

每年產出 800 頁

of dense transformative insights

密集且具有顛覆性洞見的論文，

seems to be more correlated with a flexibility towards conventions 文法

似乎更取決於對約定的靈活運用，

than it does with focusing on which standards are objectively right.

而非執著於關注哪些標準在客觀上才是絕對正確的。

So,

所以，

on this pi day,

在這個圓周率日（Pi Day），

the next time someone tells you that,

下次如果有人對你說，

you know,

你知道的，

we should really be celebrating math on June 28th,

我們其實應該在 6 月 28 日慶祝數學節，

see how quickly you can change the topic 語氣意圖

看看你能多快地把話題轉移，

to one where you're actually talking about a piece of math.

轉移到真正討論數學本身的內容上。