歷史上最奇特的數學家：英語精讀講義

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

阿基米德對他的發現感到非常興奮，以至於他光著身子跑在街上，

shouting at the top of his lungs, 慣用語

扯著嗓子大喊：

"Eureka, I have found it." 文化背景

「Eureka（找到了），我找到了。」

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

傳說敘拉古的國王——

a powerful ancient Greek city-state,

敘拉古是一個強大的古希臘城邦——

suspected that a goldsmith had cheated him

懷疑一名金匠欺騙了他，

by mixing cheaper silver into his crown.

在他的王冠裡摻了廉價的銀。

He turned to Archimedes to figure out 片語動詞

他求助於阿基米德來弄清楚

if the crown was pure gold without damaging it. 文法

在不損壞王冠的情況下，它是否為純金。

Archimedes solved the problem when, one day,

有一天，阿基米德找到了解決這個問題的方法：

he stepped into a full bathtub

他跨進一個裝滿水的浴缸，

and noticed water spilling over the edge.

並注意到水從邊緣溢了出來。

The more he immersed himself, 文法

他把自己浸得越深，

the more water overflowed.

溢出來的水就越多。

That's when he realized the amount of water displaced

就在那時他意識到，排開的水量

must be equal to the volume of the object submerged.

必定等於浸入物體的體積。

He then leaped from his bath

於是他從浴缸裡跳了出來，

and straight into the pages of history. 慣用語

並直接躍進了歷史的篇章中。

Armed with this newfound realization, 慣用語

帶著這個新發現，

he solved the king's challenge

他解決了國王的難題，

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

透過將王冠與同等重量的金塊進行比較。

When Archimedes submerged the crown,

當阿基米德將王冠浸入水中時，

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

它排開的水比同等重量的純金還要多。

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

這顯示它的體積較大，且不是純金。

It was mixed with silver.

裡面摻雜了銀。

His discovery became the foundation for understanding buoyancy, 慣用語

他的發現成為理解浮力的基礎，

the reason ships float.

這也是船隻能夠漂浮的原因。

The Roman statesman and philosopher Cicero claimed

羅馬政治家兼哲學家西塞羅聲稱，

that Archimedes possessed a genius

阿基米德擁有

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

超乎人們想像的驚人天賦。

The eureka moment was just a glimpse of his brilliance.

這個「Eureka」的時刻只是他才華的冰山一角。

Archimedes was born around 287 BCE in Syracuse,

阿基米德大約在西元前 287 年出生於敘拉古，

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

這是一個位於西西里島東南岸繁榮的希臘城邦。

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

他很幸運地出生在希倫二世統治時期，

a rare period of peace in a region

這是該地區罕見的和平時期，

otherwise marked by power struggles and invasions. 用詞

而這個地區在其他時期多半充斥著權力鬥爭與入侵。

His passion for math led him to Alexandria,

他對數學的熱愛驅使他前往亞歷山卓，

the intellectual capital of the Hellenistic world,

那裡是希臘化世界的學術之都，

where he studied with the successors of Euclid

他在那裡與歐幾里得的繼承者一起學習，

and soaked in knowledge at the great library. 慣用語

並在偉大的圖書館裡汲取知識。

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

正是在埃及，年輕的阿基米德發明了他著名的螺旋幫浦，

an ingenious device that uses a hollow spiral

這是一種巧妙的裝置，利用中空的螺旋

to lift water from rivers and wells for irrigation.

從河流和井中將水抽上來灌溉。

For thousands of years,

數千年來，

farmers relied on this simple yet effective tool

農民一直仰賴這項簡單卻有效的工具，

rooted in mathematical precision.

它植根於數學的精確性。

His passion for math would fuel a lifetime of discoveries. 慣用語

他對數學的熱情推動了他一生的發現。

He considered his greatest achievement 文法

他認為自己最大的成就是

to be uncovering a fascinating mathematical relationship

揭示了球體和圓柱體之間

between spheres and cylinders.

迷人的數學關係。

He found that the volume of a sphere

他發現球體的體積

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

是完美包圍它的圓柱體體積的三分之二，

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

並且相同的 2:3 比例也適用於它們的表面積。

Archimedes was so proud of this discovery

阿基米德對這項發現感到非常自豪，

that he requested it to be engraved on his tombstone.

甚至要求將其刻在他的墓碑上。

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

對數學的這份熱情驅使他去應對另一個永恆的挑戰，

calculating the true value of pi using only basic geometry.

僅使用基本幾何學來計算圓周率的真實數值。

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

圓周率是圓的周長與其直徑的比例。

It had already been approximated earlier

早期巴比倫人和埃及人

by the Babylonians and Egyptians.

已經得出了近似值。

Archimedes began by inscribing a triangle snugly inside the circle 文法

阿基米德開始在圓內緊密地畫一個內接三角形，

and another around it.

並在外面畫一個外切三角形。

By calculating their perimeters,

透過計算它們的周長，

he estimated that pi was between 2.60 and 5.19.

他估算圓周率介於 2.60 和 5.19 之間。

Then he used shapes that had more and more sides,

接著他使用了邊數越來越多的形狀，

more closely resembling a circle,

越來越接近圓形，

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

包括正方形、六邊形，甚至是 96 邊形。

Each step brought him closer to the value of pi.

每一步都讓他更接近圓周率的數值。

He eventually figured out it was between 3.1408 and 3.1429,

他最終計算出它介於 3.1408 和 3.1429 之間，

which is very close to the real value of pi,

這非常接近圓周率的真實數值，

about 3.14159,

大約是 3.14159，

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

確切的數值直到 18 世紀才被確定下來。

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

但阿基米德不僅僅是一位嚴肅的數學家。

He also had a playful side,

他也有俏皮的一面，

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

就像他給亞歷山卓學者出的那些腦筋急轉彎一樣。

His cattle problem involved calculating the number of cattle

他的「群牛問題」涉及計算牛的數量，

in the herd of the sun god Helios

這些牛屬於太陽神赫利俄斯的牛群，

grazing on the mythical island of Thrinacia.

在神話中的特里納基亞島上吃草。

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

阿基米德將公牛和母牛的牛群分為四組，

white, black, yellow, and dappled.

白、黑、黃和斑點。

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

接著他列出了一組必須滿足的

that had to be satisfied,

比例關係和限制條件，

resulting in a complex set of equations. 文法

形成了一組複雜的方程式。

The problem became extraordinarily difficult

這個問題變得異常困難，

when a second set of restrictions was added,

當加上第二組限制條件時，

such as requiring the total number of cattle

例如要求牛的總數

to be a triangular number.

必須是一個三角形數。

For centuries,

幾個世紀以來，

the problem remained unsolved.

這個問題一直未解。

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

直到 1880 年，德國數學家 A. Amthor 才部分解決了這個問題，

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

揭示出牛的總數是一個驚人的巨大數字。

Written out in full, 文法

如果全部寫出來，

the number would stretch to 206,545 digits.

這個數字將長達 206,545 位數。

In 1965,

1965 年，

the full problem was finally solved using IBM computers

在加拿大滑鐵盧大學，

at the University of Waterloo in Canada.

使用 IBM 電腦終於完全解決了這個問題。

Another seemingly impossible question Archimedes tackled was, 文法

阿基米德探討的另一個看似不可能的問題是，

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

要用多少粒沙子才能填滿已知宇宙？

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

這不是真的要去數沙粒，

but about demonstrating how mathematics could represent

而是為了展示數學如何能夠表達

and reason about unimaginably large numbers. 用詞

並推論難以想像的巨大數字。

The ancient Greek numeral system used letters of the alphabet

古希臘數字系統使用字母表中的字母

to represent numbers,

來代表數字，

but this system became impractical for larger numbers.

但這個系統對於較大的數字變得不實用。

Representing numbers beyond 10,000

代表超過 10,000 的數字

required increasingly complex symbols,

需要越來越複雜的符號，

which made calculations difficult.

這讓計算變得困難。

To overcome this challenge, 表達脈絡

為了克服這個挑戰，

Archimedes introduced a revolutionary system of orders of magnitude, 用詞

阿基米德引入了革命性的數量級系統，

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

對數字進行分組，使得每個數量級都比前一個大 10 的 8 次方倍。

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

他能夠表示高達 10 的 8 次方

times 10 to the power of 8.

乘以 10 的 8 次方的巨大數字。

With his innovation,

憑藉這項創新，

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

阿基米德估計需要 10 的 63 次方

grains of sand to fill the cosmos.

粒沙子才能填滿宇宙。

While modern science shows this vastly underestimates 表達脈絡

雖然現代科學表明，這大大低估了

the actual size of the universe,

宇宙的實際大小，

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

但在當時這是數學推論的突破性展現。

Archimedes shared his discoveries in unconventional letters

阿基米德透過非傳統的信件

to fellow mathematicians.

與其他數學家分享他的發現。

Alan Hirshfeld,

Alan Hirshfeld，

author of Eureka Man,

《Eureka Man,

The Life and Legacy of Archimedes,

The Life and Legacy of Archimedes》的作者，

likened them to a Hollywood movie trailer, 片語搭配

將它們比作好萊塢電影的預告片，

teasing the breakthrough and challenging his peers to solve it.

透露這些突破並挑戰他的同儕來解決。

He even included fake propositions to catch plagiarists.

他甚至加入了假的命題來抓抄襲者。

As he explained,

正如他所解釋的：

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

「那些聲稱發現了一切卻拿不出任何證明的人，

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

可能會被駁斥為假裝發現了不可能的事物。」

We may never have fully grasped Archimedes' genius 文法

我們可能永遠無法完全理解阿基米德的天才，

because one of his greatest works was nearly lost forever.

因為他最偉大的作品之一差點永遠失傳。

Archimedes recorded his ideas on papyrus scrolls,

阿基米德將他的想法記錄在莎草紙卷軸上，

the most common writing material in the ancient Greek world,

這是古希臘世界最常見的書寫材料，

but these fragile scrolls deteriorated over time.

但這些脆弱的卷軸隨著時間而毀損了。

In the 10th century,

在 10 世紀，

a Byzantine scribe copied seven of his treatises onto parchment

一位拜占庭抄寫員將他的七篇論文抄寫到羊皮紙上，

to save them from being lost forever, 文法

以免它們永遠失傳，

but by the 13th century,

但到了 13 世紀，

parchment had become so scarce

羊皮紙變得非常稀缺，

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

以至於另一位抄寫員刮掉了阿基米德的文字，

unaware of the mathematical treasure he was holding,

完全不知道他手中握著的是數學寶藏，

and reused the pages to create a Christian prayer book.

並重新使用這些書頁來製作基督教祈禱書。

Remarkably,

令人驚訝的是，

beneath the prayers,

在祈禱文的下方，

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

可以看到阿基米德作品淡淡的痕跡從左到右延伸。

In 1906,

1906 年，

the Danish scholar Johan Ludvig Heiberg

丹麥學者 Johan Ludvig Heiberg

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

發現了這份 174 頁被稱為「重寫本」的羊皮紙，

in a monastery library in Constantinople,

在君士坦丁堡的一家修道院圖書館裡，

now Istanbul.

即今日的伊斯坦堡。

Using only a magnifying glass,

僅僅使用一個放大鏡，

he uncovered the faint traces of an underlying text

他就揭露了底層文字的微弱痕跡，

and identified it as the work of Archimedes.

並確認它是阿基米德的作品。

In 1998,

1998 年，

an anonymous private American collector purchased the medieval prayer book

一位匿名的美國私人收藏家買下了這本中世紀祈禱書，

at a Christie's auction for over 2 million.

在佳士得拍賣會上以超過 200 萬美元的價格成交。

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

隨後，這位收藏家將其借給了馬里蘭州巴爾的摩的沃爾特斯美術館，

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

在那裡，它得到了仔細的保存、數位化成像，並變得可辨識。

One of the most famous ideas preserved in the manuscript

這份手稿中保存的最著名想法之一

is the law of the lever,

就是槓桿原理，

a principle that is simple yet profound.

這是一個簡單卻深奧的原理。

A small force applied at the right distance

在正確的距離施加一個小小的力量

can lift a much heavier weight.

就能舉起重得多的重量。

For example,

例如，

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

即使槓桿的一端承載著另一端兩倍的重量，

the lever will balance if the heavier side

只要較重的一端

is half as far from the fulcrum.

與支點的距離是另一端的一半，槓桿就會平衡。

Archimedes captured the power of this principle

阿基米德掌握了這個原理的力量，

when he famously said,

正如他那句名言：

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

「給我一根夠長的槓桿和一個放置它的支點，

and I shall move the world."

我就能移動地球。」

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

他對平衡和力量的理解應用於現實世界的場景，

when he famously defended Syracuse

最著名的是他在西元前 213 年保衛敘拉古，

during the siege by the Roman Republic in 213 BCE.

抵抗羅馬共和國的圍攻。

At the time,

當時，

Syracuse was aligned with Carthage,

敘拉古與迦太基結盟，

to the frustration of the Romans.

這讓羅馬人感到沮喪。

Archimedes created a crane-like mechanism resembling a claw

阿基米德創造了一種像爪子一樣的起重機裝置，

that grabbed onto Roman ships,

它能抓住羅馬的船隻，

tipping them over or lifting them out of the water

將它們弄翻，或是把它們從水中吊起，

and shaking them violently.

然後猛烈地搖晃。

The Greek philosopher and historian Plutarch describes

希臘哲學家兼歷史學家普魯塔克描述，

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

羅馬人開始認為他們是在與神作戰。

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

他的戰爭機器的巧妙設計甚至衍生出一些神話，

including one about Archimedes using mirrors

包括阿基米德使用鏡子

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

聚焦陽光來燒毀羅馬船隻的傳說。

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

更有可能的情況是，防禦者可能使用了

polished metal plates to reflect sunlight

拋光的金屬板來反射陽光，

into the eyes of Roman sailors,

照進羅馬水手的眼睛裡，

disorienting them as they approached.

在他們靠近時使他們迷失方向。

Despite Archimedes' efforts, 表達脈絡

儘管阿基米德付出了努力，

the city ultimately fell 用詞

這座城市最終還是在西元前 212 年，

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

淪陷於經驗豐富且裝備精良得多的羅馬人手中。

Archimedes was so focused on his work

阿基米德如此專注於他的工作，

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

以至於他甚至沒有意識到羅馬人已經攻破了城市。

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

傳說當時他正彎腰看著一個數學圖表，

deep in thought, 片語搭配

陷入沉思，

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

這時一名羅馬士兵走近他，並命令他離開。

Archimedes refused to abandon his calculations,

阿基米德拒絕放棄他的計算，

reportedly saying,

據說他說了：

"Do not disturb my circles,"

「不要弄亂我的圓，」

a reference to his mathematical work.

這是指他的數學研究。

Enraged,

士兵被激怒了，

the soldier killed him on the spot. 慣用語

當場殺死了他。

As author Alan Hirshfeld declared,

正如作家 Alan Hirshfeld 所宣稱的：

"He is said to have taken his dying breath 文法

「據說他在嚥下最後一口氣時，

poised between the point of a sword 慣用語

正徘徊在劍尖

and a point he was pondering in mathematics."

與他正在思考的數學論點之間。」

Archimedes was around 75 years old.

當時阿基米德大約 75 歲。

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

羅馬將軍馬塞盧斯得知阿基米德的死訊後悲痛萬分。

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

他曾特別命令他的士兵饒恕這位偉大數學家的性命，

understanding the immense value he could bring to Rome.

因為他明白他能為羅馬帶來巨大的價值。

To honor him, 表達脈絡

為了紀念他，

Marcellus arranged for a grand funeral

馬塞盧斯為他安排了一場隆重的葬禮，

and ordered a tomb inscribed with Archimedes' favorite discovery,

並下令在他的墓碑上刻上阿基米德最喜歡的發現，

the sphere and cylinder,

球體和圓柱體，

and their surface area ratio of 2 to 3.

以及它們 2 比 3 的表面積比例。

Archimedes' discoveries laid the groundwork 慣用語

阿基米德的發現奠定了基礎，

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

造就了我們今天在數學和科學中所了解的許多事物。

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

如果你想精進自己的數學能力，

Brilliant is the perfect place to start.

Brilliant 是完美的起點。

Brilliant is an interactive learning platform

Brilliant 是一個互動學習平台，

that helps you build real-world skills in math,

能幫助你透過動手解題，培養數學、

science, programming, and AI

科學、程式設計和人工智慧方面的

through hands-on problem-solving.

實用技能。

If you want to boost your understanding of geometry,

如果你想提升對幾何學的理解，

Brilliant's courses let you visualize and solve problems

Brilliant 的課程能讓你視覺化並解決問題，

in a fun and engaging way,

以一種有趣且引人入勝的方式，

or explore algebra and calculus

或是探索代數和微積分，

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

深入了解塑造我們周遭世界的公式。

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

我個人很享受每天花 10 到 15 分鐘使用 Brilliant。

Right now,

現在，

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

我正在探索他們關於大型語言模型如何運作的課程，

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

該課程分解了 ChatGPT 等人工智慧系統背後的原理，

in an easy-to-understand way.

以一種易於理解的方式呈現。

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

對任何熱愛學習的人來說，這都是一個強大的習慣。

You can try Brilliant for free for 30 days

你可以免費試用 Brilliant 30 天，

by visiting brilliant.org/newsthink.

只要造訪 brilliant.org/newsthink。

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

那就是 brilliant.org/newsthink 以獲得免費試用。

My link is in the description,

我的連結在說明欄裡，

or simply scan the QR code on your screen.

或是直接掃描你螢幕上的 QR 碼。

Plus,

此外，

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

你還可以獲得年度高級訂閱的 8 折優惠。

Thanks for watching.

感謝收看。

For NewsThink,

這裡是 NewsThink，

I'm Cindy Pom.

我是 Cindy Pom。