历史上最奇特的数学家：英语精读讲义

Archimedes was so thrilled by his discovery that he ran through the streets naked, 語法

阿基米德对自己的发现感到如此兴奋，以至于光着身子跑上街头，

shouting at the top of his lungs, 習語

扯着嗓子大喊：

"Eureka, I have found it." 文化背景

“尤里卡！我找到了！”

The story goes that the king of Syracuse, 表達脈絡

传说中，叙拉古——

a powerful ancient Greek city-state,

一个强大的古希腊城邦——的国王，

suspected that a goldsmith had cheated him

怀疑一个金匠欺骗了他，

by mixing cheaper silver into his crown.

在他的王冠里掺了廉价的银子。

He turned to Archimedes to figure out 短語動詞

他向阿基米德求助，想弄清楚

if the crown was pure gold without damaging it. 語法

这顶王冠是否是纯金的，同时又不能损坏它。

Archimedes solved the problem when, one day,

阿基米德解决这个问题的契机是，有一天，

he stepped into a full bathtub

他迈进装满水的浴缸，

and noticed water spilling over the edge.

注意到水溢出了边缘。

The more he immersed himself, 語法

他浸入水中的身体越多，

the more water overflowed.

溢出的水就越多。

That's when he realized the amount of water displaced

就在那时他意识到，排开的水量

must be equal to the volume of the object submerged.

必然等于沉入水中的物体的体积。

He then leaped from his bath

然后他从浴缸里跳了出来，

and straight into the pages of history. 習語

直接跃入了史册。

Armed with this newfound realization, 習語

带着这个新发现，

he solved the king's challenge

他解决了国王的难题，

by comparing the crown to a gold bar of equal weight. 語法

将王冠与同等重量的金块进行了比较。

When Archimedes submerged the crown,

当阿基米德将王冠浸入水中时，

it displaced more water than pure gold of the same weight.

它排开的水比同等重量的纯金更多。

This revealed it had a larger volume and wasn't pure gold.

这表明它的体积更大，因此不是纯金。

It was mixed with silver.

里面掺了银子。

His discovery became the foundation for understanding buoyancy, 習語

他的发现成为了理解浮力的基础，

the reason ships float.

这也是船只能够漂浮的原因。

The Roman statesman and philosopher Cicero claimed

罗马政治家和哲学家西塞罗声称，

that Archimedes possessed a genius

阿基米德拥有常人难以想象的

greater than one would imagine it possible for a human being. 語法

超凡天赋。

The eureka moment was just a glimpse of his brilliance.

“尤里卡”时刻仅仅是他才华的冰山一角。

Archimedes was born around 287 BCE in Syracuse,

阿基米德约在公元前287年出生于叙拉古，

a prosperous Greek city-state on the southeastern shore of Sicily.

这是西西里岛东南岸一个繁荣的希腊城邦。

He was fortunate to be born during the reign of Hieron II,

他很幸运地出生在希伦二世统治时期，

a rare period of peace in a region

这是该地区罕见的和平时期，

otherwise marked by power struggles and invasions. 用詞

而其他时候则充满了权力斗争和入侵。

His passion for math led him to Alexandria,

对数学的热爱驱使他前往亚历山大，

the intellectual capital of the Hellenistic world,

那是希腊化世界的学术之都，

where he studied with the successors of Euclid

在那里他与欧几里得的后继者们一起学习，

and soaked in knowledge at the great library. 習語

并在大图书馆里汲取知识。

It was in Egypt that the young Archimedes invented his famous screw, 語法

正是在埃及，年轻的阿基米德发明了著名的阿基米德螺旋泵，

an ingenious device that uses a hollow spiral

这是一种巧妙的装置，利用中空的螺旋管

to lift water from rivers and wells for irrigation.

将水从河流和井中抽出用于灌溉。

For thousands of years,

几千年来，

farmers relied on this simple yet effective tool

农民们一直依赖这种简单却有效的工具，

rooted in mathematical precision.

其核心在于数学上的精确。

His passion for math would fuel a lifetime of discoveries. 習語

他对数学的热情将推动他一生的发现。

He considered his greatest achievement 語法

他认为自己最伟大的成就

to be uncovering a fascinating mathematical relationship

是揭示了一个引人入胜的数学关系——

between spheres and cylinders.

球体和圆柱体之间的关系。

He found that the volume of a sphere

他发现球体的体积

is two-thirds the volume of a cylinder that perfectly surrounds it,

是完美包围它的外接圆柱体体积的三分之二，

and the same 2:3 ratio applies to their surface areas.

同样2:3的比例也适用于它们的表面积。

Archimedes was so proud of this discovery

阿基米德对这项发现感到非常自豪，

that he requested it to be engraved on his tombstone.

以至于他要求将其刻在自己的墓碑上。

That passion for math led him to tackle another timeless challenge,

这种对数学的热情促使他去应对另一个永恒的挑战：

calculating the true value of pi using only basic geometry.

仅用基本几何学来计算圆周率的真实数值。

Pi is the ratio of a circle's circumference to its diameter. 用詞

圆周率是圆的周长与其直径的比值。

It had already been approximated earlier

在更早的时候，它已被估算过，

by the Babylonians and Egyptians.

由巴比伦人和埃及人。

Archimedes began by inscribing a triangle snugly inside the circle 語法

阿基米德首先在圆内紧凑地画了一个内接三角形，

and another around it.

在圆外画了一个外接三角形。

By calculating their perimeters,

通过计算它们的周长，

he estimated that pi was between 2.60 and 5.19.

他估计圆周率在2.60和5.19之间。

Then he used shapes that had more and more sides,

然后他使用了边数越来越多的形状，

more closely resembling a circle,

使其更接近一个圆形，

squares, hexagons, and even a 96-sided polygon.

正方形、六边形，甚至96边形。

Each step brought him closer to the value of pi.

每一步都让他更接近圆周率的值。

He eventually figured out it was between 3.1408 and 3.1429,

最终他计算出它介于3.1408和3.1429之间，

which is very close to the real value of pi,

这非常接近真实的圆周率，

about 3.14159,

大约3.14159，

which wasn't determined with certainty until the 18th century. 語法

而确切的值直到18世纪才被确定下来。

But Archimedes wasn't just a serious mathematician. 表達脈絡

但阿基米德不仅仅是一位严肃的数学家。

He also had a playful side,

他也有爱玩的一面，

as seen in the brain teasers he posed to scholars in Alexandria. 用詞

比如他给亚历山大的学者们出的脑筋急转弯。

His cattle problem involved calculating the number of cattle

他的“群牛问题”涉及计算太阳神赫利俄斯

in the herd of the sun god Helios

在神话岛屿斯里纳基亚上放牧的

grazing on the mythical island of Thrinacia.

牛群数量。

Archimedes divided the herd of bulls and cows into four groups,

阿基米德将公牛和母牛分为四组：

white, black, yellow, and dappled.

白牛、黑牛、黄牛和斑点牛。

He then laid out a set of proportional relationships and constraints 短語動詞

然后他提出了一组比例关系和限制条件，

that had to be satisfied,

这必须被满足，

resulting in a complex set of equations. 語法

从而得到了一组复杂的方程式。

The problem became extraordinarily difficult

这个问题变得异常困难，

when a second set of restrictions was added,

当加入第二组限制条件时，

such as requiring the total number of cattle

比如要求牛的总数

to be a triangular number.

必须是一个三角形数。

For centuries,

几个世纪以来，

the problem remained unsolved.

这个问题一直悬而未决。

It wasn't until 1880 that German mathematician A. Amthor partially solved it, 語法

直到1880年，德国数学家A·阿姆托尔才部分解决它，

revealing that the total number of cattle was an alarmingly large number.

揭示出牛的总数是一个惊人的庞大数字。

Written out in full, 語法

如果完整写出来，

the number would stretch to 206,545 digits.

这个数字将长达206,545位。

In 1965,

在1965年，

the full problem was finally solved using IBM computers

完整的问题终于利用IBM计算机

at the University of Waterloo in Canada.

在加拿大的滑铁卢大学被解决了。

Another seemingly impossible question Archimedes tackled was, 語法

阿基米德应对的另一个看似不可能的问题是：

how many grains of sand would it take to fill the known universe?

填满已知宇宙需要多少粒沙子？

This wasn't about literally counting grains of sand, 表達脈絡

这并非字面上去数沙粒，

but about demonstrating how mathematics could represent

而是为了证明数学可以如何表示

and reason about unimaginably large numbers. 用詞

并推导难以想象的巨大数字。

The ancient Greek numeral system used letters of the alphabet

古希腊的数字系统使用字母

to represent numbers,

来表示数字，

but this system became impractical for larger numbers.

但这种系统对于更大的数字变得不切实际。

Representing numbers beyond 10,000

表示超过10,000的数字

required increasingly complex symbols,

需要越来越复杂的符号，

which made calculations difficult.

这使得计算变得困难。

To overcome this challenge, 表達脈絡

为了克服这一挑战，

Archimedes introduced a revolutionary system of orders of magnitude, 用詞

阿基米德引入了一种革命性的数量级系统，

grouping numbers so that each order was 10 to the power of 8 times larger than the one before.

将数字分组，使每个数量级比前一个大10的8次方倍。

He could represent numbers as large as 10 raised to the power of 8

他能够表示大到10的8次方

times 10 to the power of 8.

乘以10的8次方的数字。

With his innovation,

凭借这项创新，

Archimedes estimated that it would take 10 to the power of 63

阿基米德估计需要10的63次方

grains of sand to fill the cosmos.

粒沙子才能填满宇宙。

While modern science shows this vastly underestimates 表達脈絡

虽然现代科学表明这极大地低估了

the actual size of the universe,

宇宙的实际大小，

it was a groundbreaking demonstration of mathematical reasoning at the time.

但在当时，这已经是数学推理的开创性展示。

Archimedes shared his discoveries in unconventional letters

阿基米德通过不同寻常的信件分享他的发现，

to fellow mathematicians.

寄给其他数学家同行。

Alan Hirshfeld,

艾伦·赫什菲尔德，

author of Eureka Man,

《尤里卡之人》一书的作者，

The Life and Legacy of Archimedes,

副标题为《阿基米德的生平与遗产》，

likened them to a Hollywood movie trailer, 短語搭配

将其比作好莱坞电影预告片，

teasing the breakthrough and challenging his peers to solve it.

先吊人胃口地透露突破，并挑战同行去求解。

He even included fake propositions to catch plagiarists.

他甚至加入了虚假的命题以揪出剽窃者。

As he explained,

正如他所解释的：

"Those who claim to discover everything but produce no proofs of the same

“那些声称发现了一切却没有提供证明的人，

may be confuted as having pretended to discover the impossible."

可能会被拆穿是在假装发现了不可能的事情。”

We may never have fully grasped Archimedes' genius 語法

我们可能永远无法完全领会阿基米德的天才，

because one of his greatest works was nearly lost forever.

因为他的一部最伟大的作品险些永远失传。

Archimedes recorded his ideas on papyrus scrolls,

阿基米德将他的思想记录在莎草纸卷轴上，

the most common writing material in the ancient Greek world,

这是古希腊世界最常见的书写材料，

but these fragile scrolls deteriorated over time.

但这些脆弱的卷轴随着时间的推移而腐烂。

In the 10th century,

在10世纪，

a Byzantine scribe copied seven of his treatises onto parchment

一位拜占庭抄写员将他的七部论文抄写到羊皮纸上，

to save them from being lost forever, 語法

以防止它们永远遗失，

but by the 13th century,

但到了13世纪，

parchment had become so scarce

羊皮纸变得如此稀缺，

that another scribe scraped off Archimedes' text, 短語動詞

以至于另一位抄写员刮掉了阿基米德的文字，

unaware of the mathematical treasure he was holding,

对这份数学瑰宝毫无察觉，

and reused the pages to create a Christian prayer book.

并重复利用这些书页制作了一本基督教祈祷书。

Remarkably,

奇迹般地，

beneath the prayers,

在祈祷文之下，

faint traces of Archimedes' work could be seen running from left to right.

仍能看到从左到右延伸的阿基米德作品的淡淡痕迹。

In 1906,

在1906年，

the Danish scholar Johan Ludvig Heiberg

丹麦学者约翰·路德维格·海堡

discovered the 174-page parchment called the palimpsest 文化背景

发现了这份被称为“重写本”的174页羊皮纸，

in a monastery library in Constantinople,

在君士坦丁堡的一家修道院图书馆里，

now Istanbul.

也就是现在的伊斯坦布尔。

Using only a magnifying glass,

仅用一个放大镜，

he uncovered the faint traces of an underlying text

他发现了底层文字的微弱痕迹，

and identified it as the work of Archimedes.

并确认这是阿基米德的作品。

In 1998,

1998年，

an anonymous private American collector purchased the medieval prayer book

一位匿名的美国私人收藏家买下了这本中世纪祈祷书，

at a Christie's auction for over 2 million.

在佳士得拍卖会上以超过200万美元的价格。

The collector then loaned it to the Walters Art Museum in Baltimore, Maryland,

随后收藏家将其借给马里兰州巴尔的摩的沃尔特斯美术馆，

where it was carefully conserved, digitally imaged, and made legible. 語法

在那里它被小心地保存、进行数字成像并使其可读。

One of the most famous ideas preserved in the manuscript

手稿中保存的最著名的理念之一

is the law of the lever,

是杠杆定律，

a principle that is simple yet profound.

一个简单却深奥的原理。

A small force applied at the right distance

在合适的距离施加很小的力量，

can lift a much heavier weight.

就可以举起重得多的物体。

For example,

例如，

even if one side of the lever carries twice the weight of the other, 語法

即使杠杆一侧承载的重量是另一侧的两倍，

the lever will balance if the heavier side

只要较重一侧距离支点

is half as far from the fulcrum.

只有另一侧的一半远，杠杆就会平衡。

Archimedes captured the power of this principle

阿基米德用一句名言捕捉了这个原理的威力：

when he famously said,

当他留下那句著名的话，

"Give me a lever long enough and a fulcrum on which to place it, 表達脈絡

“给我一个足够长的杠杆和一个放置它的支点，

and I shall move the world."

我就能撬动地球。”

His understanding of balance and force applied to real-world scenarios,

他对平衡和力的理解被应用于现实场景，

when he famously defended Syracuse

在他著名的保卫叙拉古之战中，

during the siege by the Roman Republic in 213 BCE.

在公元前213年被罗马共和国围攻期间。

At the time,

当时，

Syracuse was aligned with Carthage,

叙拉古与迦太基结盟，

to the frustration of the Romans.

这令罗马人非常沮丧。

Archimedes created a crane-like mechanism resembling a claw

阿基米德发明了一种类似起重机的机械爪，

that grabbed onto Roman ships,

能够抓住罗马的船只，

tipping them over or lifting them out of the water

将它们翻转或从水里拉出来，

and shaking them violently.

并剧烈地摇晃它们。

The Greek philosopher and historian Plutarch describes

希腊哲学家兼历史学家普鲁塔克描述道：

how the Romans began to think they were fighting with the gods.

罗马人开始觉得他们是在与神明作战。

The ingenuity of his war machines even gave rise to myths, 短語搭配

他那些战争机器的巧妙设计甚至催生了神话，

including one about Archimedes using mirrors

其中包括阿基米德利用镜子

to focus the sun's rays and burn Roman ships.

聚焦太阳光线来烧毁罗马战船的传说。

What's more likely is that the defenders may have used 講解

更可能的情况是，防御者们使用了

polished metal plates to reflect sunlight

抛光的金属板来反射阳光

into the eyes of Roman sailors,

直射罗马水手的眼睛，

disorienting them as they approached.

使他们在靠近时迷失方向。

Despite Archimedes' efforts, 表達脈絡

尽管阿基米德付出了努力，

the city ultimately fell 用詞

这座城市最终还是沦陷了，

to the vastly more experienced and better-equipped Romans in 212 BCE.

败给了经验丰富得多且装备精良的罗马人，时值公元前212年。

Archimedes was so focused on his work

阿基米德极其专注于他的工作，

that he didn't even realize that the Romans had stormed the city.

甚至没有意识到罗马人已经攻陷了这座城市。

Legend has it that he was hunched over a mathematical diagram, 講解

传说他正躬身于一个数学图表，

deep in thought, 短語搭配

陷入沉思，

when a Roman soldier approached him and ordered him to leave.

这时一名罗马士兵走近他并命令他离开。

Archimedes refused to abandon his calculations,

阿基米德拒绝放弃他的计算，

reportedly saying,

据报道他说，

"Do not disturb my circles,"

“别弄坏我的圆，”

a reference to his mathematical work.

这是指他的数学工作。

Enraged,

士兵暴怒之下，

the soldier killed him on the spot. 習語

当场杀死了他。

As author Alan Hirshfeld declared,

正如作家艾伦·赫什菲尔德所称：

"He is said to have taken his dying breath 語法

“据说他咽下最后一口气时，

poised between the point of a sword 習語

悬于剑尖

and a point he was pondering in mathematics."

与他正在思索的数学中的一个点之间。”

Archimedes was around 75 years old.

阿基米德当时大约75岁。

The Roman general Marcellus was devastated to learn of Archimedes' death.

罗马将军马塞拉斯得知阿基米德的死讯后悲痛万分。

He had specifically ordered his soldiers to spare the life of the great mathematician, 短語搭配

他曾特意下令他的士兵饶这位伟大数学家一命，

understanding the immense value he could bring to Rome.

因为他明白阿基米德能给罗马带来巨大的价值。

To honor him, 表達脈絡

为了向他致敬，

Marcellus arranged for a grand funeral

马塞拉斯为他安排了一场盛大的葬礼，

and ordered a tomb inscribed with Archimedes' favorite discovery,

并下令在他的墓碑上刻上阿基米德最喜爱的发现，

the sphere and cylinder,

即球体和圆柱体，

and their surface area ratio of 2 to 3.

以及它们2比3的表面积比例。

Archimedes' discoveries laid the groundwork 習語

阿基米德的发现奠定了基础，

for so much of what we understand in math and science today.

造就了我们今天在数学和科学中所理解的许多东西。

If you'd like to sharpen your own math skills, 習語

如果你想提高自己的数学技能，

Brilliant is the perfect place to start.

Brilliant 是一个完美的起点。

Brilliant is an interactive learning platform

Brilliant 是一个互动学习平台，

that helps you build real-world skills in math,

帮助你在数学方面构建现实世界的技能，

science, programming, and AI

科学、编程和人工智能，

through hands-on problem-solving.

通过动手解决问题。

If you want to boost your understanding of geometry,

如果你想提升对几何学的理解，

Brilliant's courses let you visualize and solve problems

Brilliant的课程能让你可视化并解决问题，

in a fun and engaging way,

以一种有趣且引人入胜的方式，

or explore algebra and calculus

或者探索代数和微积分，

to dive deeper into the formulas that shape the world around us. 短語動詞

以更深入地研究塑造我们周围世界的公式。

I personally enjoy spending 10 to 15 minutes a day with Brilliant.

我个人很喜欢每天在 Brilliant 上花10到15分钟。

Right now,

现在，

I'm exploring their how large language models work course,

我正在探索他们关于大型语言模型如何运作的课程，

which breaks down the principles behind AI systems like ChatGPT 短語動詞

它解析了ChatGPT等AI系统背后的原理，

in an easy-to-understand way.

以一种易于理解的方式。

It's a powerful habit for anyone who loves learning.

对于任何热爱学习的人来说，这是一个强大的习惯。

You can try Brilliant for free for 30 days

你可以免费试用 Brilliant 30天，

by visiting brilliant.org/newsthink.

只需访问 brilliant.org/newsthink。

That's brilliant.org/newsthink for a free trial.

也就是 brilliant.org/newsthink 来获取免费试用。

My link is in the description,

我的链接在描述中，

or simply scan the QR code on your screen.

或者直接扫描屏幕上的二维码。

Plus,

此外，

you'll get 20% off an annual premium subscription.

你将获得年度高级订阅的20%折扣。

Thanks for watching.

感谢收看。

For NewsThink,

这里是 NewsThink，

I'm Cindy Pom.

我是 Cindy Pom。