为什么圆周率差点成了6.283185...：英语精读讲义

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

相信大家对 pi 和 tau 的争论已经不陌生了。

A lot of people say that the fundamental circle constant

许多人认为，作为基础的圆常数，

we hold up should be the ratio 短語動詞

我们所推崇的应该是

of a circle's circumference to its radius,

圆的周长与其半径的比值，

which is around 6.28,

也就是大约 6.28，

not the ratio to its diameter,

而不是周长与直径的比值，

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，我们常把那个更大的常数称为 tau，

popularized by Michael Hartl's Tau Manifesto.

这一称呼由 Michael Hartl 的《Tau 宣言》所普及。

Although personally,

虽然就我个人而言，

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

我相当偏爱 Robert Palais 提议的符号，

of a pi with three legs.

也就是长了三条腿的 pi。

In either of these manifestos

无论是这两份宣言中，

and on many many other places of the internet,

还是互联网上的许多其他地方，

you can read to no end 習語

你都可以读到数不胜数的文章，

about how many formulas look a lot cleaner using tau,

讲述使用 tau 能让多少公式看起来简洁得多，

largely because the number of radians

很大程度上是因为，

describing a given fraction of a circle

在描述圆中特定比例的弧度数值时，

is actually that fraction of tau.

恰好就是 tau 的该等比例。

That dead horse is beat. 習語

这个话题已经被说烂了。

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

我今天不是来继续论证这一点的。

Instead,

相反，

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

我想谈谈历史上的那个关键时刻：

when pi as we know it became the standard.

也就是我们所熟知的 pi 成为标准常数的时候。

For this,

为此，

one fruitful place to look

一个非常值得探索的方向，

is at the old notes and letters

就是去查阅那些旧笔记和信件，

by one of history's most influential mathematicians,

它们出自历史上最具影响力的数学家之一：

Leonhard Euler.

莱昂哈德·欧拉（Leonhard Euler）。

Luckily, we now have an official

幸运的是，我们现在有一位官方的

3Blue1Brown Switzerland correspondent, Ben Hambrecht,

3Blue1Brown 驻瑞士记者 Ben Hambrecht，

who was able to go to the library

他有幸前往

in Euler's hometown

欧拉家乡的图书馆，

and get his hands on some of the original documents. 習語

查阅到了一些原始文献。

And in looking through some of those,

在翻阅其中的部分文献时，

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

你可能会惊讶地发现欧拉写道：

"Let pi be the circumference of a circle

“令 pi 为一个圆的周长，

whose radius is one."

其半径为 1。”

That is the 6.28 constant

也就是说，这个 6.28 常数

that we would now call tau.

就是我们现在所说的 tau。

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

而且他当时很可能是将希腊字母 pi

as a P for perimeter.

作为周长（perimeter）单词首字母 P 来使用的。

So was it the case that Euler,

那么情况是否真的是，

genius of the day,

作为当代天才的欧拉，

was more notationally enlightened than the rest of the world,

在符号学上比世界上其他人更加开明，

fighting the good fight for 6.28? 習語

为推广 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

那么我们这个故事里的反派是谁呢？

pushing the 3.1415 constant

是谁在推行 3.1415 这个常数，

shoved in front of most students today?

并将其强塞给如今的大多数学生呢？

Well,

好吧，

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

真正将我们现在所知的 pi

as the commonly recognized circle constant

确立为公认圆常数的著作，

was an early calculus book from 1748.

是 1748 年出版的一本早期微积分教材。

At the start of chapter 8,

在第 8 章的开头，

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

在描述半径为 1 的圆的半周长时，

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

并且在将这个数整整展开了 128 位数字之后，

one of them wrong by the way,

（顺便一提，其中有一位写错了），

the author adds,

作者补充道：

"Which for the sake of brevity, 短語搭配

“为了简便起见，

I may write pi."

我可以将其写为 pi。”

Now, there were other texts and letters here and there

诚然，当时各处还有其他的文献和信件，

with varying conventions for the notation

对于各种与圆有关的常数的符号

of various circle constants,

有着不同的使用习惯，

but this book and this section in particular

但正是这本书，尤其是这一节，

was really the one to spread the notation throughout Europe 語法

真正将这个符号传播到了整个欧洲，

and eventually the world.

并最终传遍了全世界。

So,

那么，

what monster wrote this book 語氣意圖

究竟是哪个“怪物”写了这本书，

with such an unprincipled take towards circle constants?

对圆常数采取如此毫无原则的态度？

Well,

嗯，

Euler again.

还是欧拉。

In fact,

事实上，

if you look further,

如果你进一步挖掘，

you can find instances of Euler using the symbol pi

你还会发现欧拉使用符号 pi

to represent a quarter turn of the circle,

来表示四分之一圆周的实例，

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

也就是我们今天所说的二分之 pi 或四分之 tau。

In fact, Euler's use of the letter pi

事实上，欧拉对字母 pi 的使用

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

似乎更类似于我们对希腊字母 theta 的使用。

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

我们通常让它代表一个角度，

but no one angle in particular.

但并不特指某一个特定的角度。

Sometimes it's 30 degrees,

有时它是 30 度，

maybe other times it's 135,

也许其他时候是 135 度，

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

大多数时候它只是用于一般性陈述的变量。

It depends on the problem and the context before us.

这取决于我们面前的问题和语境。

Likewise, Euler let pi represent whatever circle constant

同样地，欧拉让 pi 代表任何

best suited the problem before him.

最适合他眼前问题的圆常数。

Though it's worth pointing out 表達脈絡

不过值得指出的是，

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

他通常是以半径为 1 的单位圆为基础来构建理论的，

so the 3.1415 constant

所以 3.1415 这个常数

would almost always have been thought of as the ratio 語法

几乎总是被认为代表了

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

圆的半周长与半径的比值，

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

而绝不是周长与直径比值这种无聊的东西。

And I think Euler's use of this symbol

我认为欧拉对这个符号的使用

carries with it a general lesson 習語

给我们带来了一个普适的启示，

about how we should approach math.

关于我们应该如何对待数学。

The thing you have to understand about Euler 語法

关于欧拉，你必须了解的一点是，

is that this man solved problems,

这个人一直在解决问题，

a lot of problems.

解决大量的问题。

I mean,

我的意思是，

day in day out, 習語

日复一日，

breakfast, lunch, and dinner,

不论是早饭、午饭还是晚饭时间，

he was just churning out puzzles and formulas 短語動詞

他总是在不断产出各种难题和公式，

and having insights and creating entire new fields left and right. 習語

并接连获得洞见，随手开创出一个个全新的领域。

Over the course of his life,

在他的一生中，

he wrote over 500 books and papers,

他写了超过 500 本书和论文，

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

算下来以每年 800 页的速度产出。

And these are dense math pages.

而且这些都是信息密集的数学文献。

And then,

此外，

after his death,

在他去世后，

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

人们经常开玩笑说，数学中的公式

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

必须以第二个证明它的人来命名，

because the first is always going to be Euler.

因为第一个证明的人总是欧拉。

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

他的心思并没有集中在

we should take as fundamental.

究竟该把哪个圆常数视作最基础的问题上。

It was on solving the task sitting in front of him

他关注的是解决在那一刻

in a particular moment

正摆在他面前的具体任务，

and writing a letter to the Bernoullis

然后在事后给伯努利家族写封信

to boast about doing so afterwards.

去吹嘘自己搞定了。

For some problems, 表達脈絡

对于某些问题，

the quarter circle constant was most natural to think about.

采用四分之一圆常数是最自然的思考方式。

For others,

对于其他问题，

the full circle constant.

则是全圆常数。

And for others still,

而对于还有些问题，

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

比如在他著名微积分教材第 8 章的开头，

maybe the half circle constant

也许半圆常数

was most natural to think about.

才是最自然的思考方式。

Too often in math education,

在数学教育中，往往过于

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

把焦点放在关于某个主题的多个相互竞争的观点中

about a topic is right.

究竟哪一个是正确的。

Is it correct to say

我们应该说

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

所有正整数之和为 -1/12 是正确的吗？

Or is it correct to say

还是应该说

that it diverges to infinity?

它发散到无穷大才是正确的？

Can the infinitesimal values of calculus

微积分中的无穷小量

be taken literally, 短語搭配

可以从字面上来理解吗，

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

还是说只能用极限的语言来表述才正确？

Are you allowed to divide a number by zero?

你能够允许把一个数除以零吗？

These questions in isolation 短語搭配

孤立地看待这些问题，

just don't matter.

其实根本无关紧要。

Our focus should be on specific problems and puzzles,

我们的重点应该放在具体的问题和谜题上，

both those of practical application

既包括那些有实际应用价值的，

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

也包括那些纯粹为了求知而进行的悠闲思考。

Then,

那么，

when questions of standards arise,

当标准问题出现时，

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

你就可以根据给定的语境来回答它们了。

And inevitably,

并且不可避免地，

different contexts will lend themselves 短語搭配

不同的语境自然会

to different answers

引导出不同的答案，

of what seems most natural.

以最自然的方式呈现。

But that's okay.

但这没关系。

Outputting 800 pages a year

每年产出 800 页

of dense transformative insights

密集且具有颠覆性洞见的论文，

seems to be more correlated with a flexibility towards conventions 語法

似乎更取决于对约定的灵活运用，

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

而非执着于关注哪些标准在客观上才是绝对正确的。

So,

所以，

on this pi day,

在这个圆周率日（Pi Day），

the next time someone tells you that,

下次如果有人对你说，

you know,

你知道的，

we should really be celebrating math on June 28th,

我们其实应该在 6 月 28 日庆祝数学节，

see how quickly you can change the topic 語氣意圖

看看你能多快地把话题转移，

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

转移到真正讨论数学本身的内容上。