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mc1-019 · ENGLISH LEARNING

幻方Magic Squares

英语学习拓展 约 10 分钟阅读 中文English

内容概述

  • 数学故事
  • 文物起源 / 数学史
  • 叙事 + 图注

本文介绍中国最古老的数学经典之一《易经》中出现的"洛书"——已知最古老的幻方。传说公元前约 2200 年,大禹在黄河之滨发现神龟背负此图。洛书是一个三阶方阵(3×3),以点表示数字(黑点为偶数、白点为奇数),其特点是任意一行、任意一列及两条对角线上的三数之和均相等。文章同时提及秦始皇焚书事件对考证中国上古数学文献年代造成的困难。

中文正文

19° 幻方

中国的数学作品,至少其中的某些部分,也可以追溯到久远的年代。这一点不容易证实,因为我们缺乏原始材料。公元前 213 年,秦始皇下令焚书坑儒,尽管命令不可能完全执行,烧毁的图书也可能凭记忆保存下来,但我们今天还是怀疑那些东西真是在那个不幸的年代之前出现的。

最古老的中国数学经典之一是《易经》。书中出现一张数字图,即著名的《洛书》(图 2)。

图 2 洛书
图 2 洛书

图 2 洛书。相传大禹治水时,一神龟从洛河爬出,背上数字排列为"数九宫",左三右七,二四为肩,六八为足,五居中央,这就是洛书。

洛书是最古老的幻方,传说,公元前 2200 年,大禹在黄河之滨发现了它,原来刻画在一只神龟的背上。如图 2 画的,它是四方排列的数字图,数字用点表示,黑点代表偶数,白点代表奇数。任意一行的三个数、任意一列的三个数,以及每条对角线的三个数,它们的和都是相等的。

英文正文

19° Magic squares. There are some Chinese mathematical works of which parts, at least, are claimed to date from very early times. This is difficult to verify because we lack original sources. As an added complication it was decreed, in 213 b.c. by the Emperor Shi Huang-ti, that all books in the country be burned and all scholars be buried. Although the edict was most certainly not completely carried out, and many books that were burned were soon restored from memory, we are now in doubt as to the genuineness of anything claimed to be older than the unfortunate date.

One of the oldest of the Chinese mathematical classics is the I-king, or Book on Permutations. In this appears a numbered diagram, known as the lo-shu, later pictured as in Figure 2.

The lo-shu is the oldest known example of a magic square, and myth claims that it was first seen by the Emperor Yu, in about 2200 b.c., as a decoration upon the back of a divine tortoise along a bank of the Yellow River. It is a square array of numerals indicated in Figure 2 by knots in strings, black knots for even numbers and white knots for odd numbers. The sum of the three numbers in any column is equal to the sum of the three numbers in any row, and also to the sum of the three numbers in either diagonal.

英语学习拓展

常用词汇

重点 · 近义词拓展
verify

音标/ˈvɛrɪfaɪ/

释义与用法

证实,核实。

原文用法:"This is difficult to verify because we lack original sources."

查看对应近义词组 ↓
decreed

音标/dɪˈkriːd/

释义与用法

颁布(法令)。

原文用法:"As an added complication it was decreed, in 213 b.c. by the Emperor Shi Huang-ti..."

edict

音标/ˈiːdɪkt/

释义与用法

法令,诏令。

原文用法:"Although the edict was most certainly not completely carried out..."

重点 · 近义词拓展
genuineness

音标/ˈdʒɛnjuɪnnəs/

释义与用法

真实性。

原文用法:"...we are now in doubt as to the genuineness of anything claimed to be older..."

查看对应近义词组 ↓
permutation

音标/ˌpɜːrmjuˈteɪʃn/

释义与用法

排列;变换。

原文用法:"the I-king, or Book on Permutations."

重点 · 近义词拓展
array

音标/əˈreɪ/

释义与用法

阵列,数组。

原文用法:"It is a square array of numerals indicated in Figure 2..."

查看对应近义词组 ↓
numeral

音标/ˈnuːmərəl/

释义与用法

数字。

原文用法:"It is a square array of numerals indicated in Figure 2..."

divine

音标/dɪˈvaɪn/

释义与用法

神圣的。

原文用法:"...as a decoration upon the back of a divine tortoise along a bank of the Yellow River."

tortoise

音标/ˈtɔːrtəs/

释义与用法

龟。

原文用法:"...upon the back of a divine tortoise along a bank of the Yellow River."

diagonal

音标/daɪˈæɡənl/

释义与用法

对角线。

原文用法:"...and also to the sum of the three numbers in either diagonal."

同义词与近义词

verify/confirm/validate

对应核心词汇:verify

用法辨析

对应核心词汇:verify

辨析:verify 强调"检验真伪";confirm 强调"确认已有信念";validate 强调"使正式有效/认可"。

decree/order/proclaim
用法辨析

对应核心词汇:decreed

辨析:decree 最正式,多用于君主或权威机构;order 较通用;proclaim 侧重"公开宣布"。

genuineness/authenticity/validity

对应核心词汇:genuineness

用法辨析

对应核心词汇:genuineness

辨析:genuineness 侧重"非伪造";authenticity 侧重"来源可靠";validity 侧重"逻辑或法律上的有效性"。

array/matrix/grid

对应核心词汇:array

用法辨析

对应核心词汇:array

辨析:array 是通用术语;matrix 特指数学中的矩阵(可进行运算);grid 侧重"网格"的视觉形象。

常用短语与固定搭配

  • date from:追溯到,始于。原文用法:"...are claimed to date from very early times."
  • as an added complication:作为额外的复杂因素。原文用法:"As an added complication it was decreed..."
  • carry out:执行,实施。原文用法:"Although the edict was most certainly not completely carried out..."
  • restore from memory:凭记忆恢复/重写。原文用法:"...many books that were burned were soon restored from memory..."
  • in doubt as to:对……存有疑问。原文用法:"we are now in doubt as to the genuineness of anything..."
  • known as:被称为,以……著称。原文用法:"...known as the lo-shu, later pictured as in Figure 2."

常用句式

  • There are some ... of which parts, at least, are claimed to...:有些……其中至少有一部分被宣称……(表达谨慎的历史断言)。原文例句:"There are some Chinese mathematical works of which parts, at least, are claimed to date from very early times."
  • It was decreed that...:(某人)颁布法令规定……(被动语态表述官方行为)。原文例句:"it was decreed, in 213 b.c. by the Emperor Shi Huang-ti, that all books in the country be burned..."
  • Although ..., ...:虽然……,但是……(让步状语从句,常用于修正极端说法)。原文例句:"Although the edict was most certainly not completely carried out, and many books that were burned were soon restored from memory, we are now in doubt as to..."

长难句解析

"Although the edict was most certainly not completely carried out, and many books that were burned were soon restored from memory, we are now in doubt as to the genuineness of anything claimed to be older than the unfortunate date."

  • 主干:we are now in doubt(我们现在存有疑问)
  • 修饰与从句:

- "Although the edict was most certainly not completely carried out," —— 让步状语从句 1 - "and many books that were burned were soon restored from memory," —— 并列的让步状语从句 2(内含定语从句 "that were burned" 修饰 "many books") - "as to the genuineness of anything claimed to be older than the unfortunate date" —— 介词短语作状语,说明 doubt 的内容 - "claimed to be older than..." —— 过去分词短语作后置定语,修饰 "anything"

  • 逻辑关系:让步(Although A and B)→ 结论(但仍 C),体现学术写作中"承认反例但不改变结论"的严谨态度。

"The lo-shu is the oldest known example of a magic square, and myth claims that it was first seen by the Emperor Yu, in about 2200 b.c., as a decoration upon the back of a divine tortoise along a bank of the Yellow River."

  • 主干:The lo-shu is the oldest known example... and myth claims that...(并列句:事实陈述 + 传说引述)
  • 修饰与从句:

- "of a magic square" —— 介词短语修饰 example - "that it was first seen by the Emperor Yu..." —— 宾语从句,说明 claims 的内容 - "in about 2200 b.c." —— 时间插入语 - "as a decoration upon the back of a divine tortoise along a bank of the Yellow River" —— 方式/地点状语,修饰 seen

  • 逻辑关系:并列(事实 + 传说),作者用 "and myth claims that" 明确将后一半标记为传说而非史实,体现了审慎的学术态度。

语言使用特点

  • 本文采用历史叙述(historical narrative)审慎推断(cautious inference) 相结合的风格。频繁使用 "claimed to", "myth claims", "it was decreed", "most certainly not" 等表达来标记信息的确信程度。
  • 被动语态密集("was decreed", "be burned", "be buried", "was carried out", "were restored", "is known", "was first seen"),这是学术英语的典型特征,用于弱化施动者、突出事件本身。
  • 专有名词处理:I-kinglo-shu 使用斜体表明其为外来词/书名;Shi Huang-ti、Yu、Yellow River 等保留英文拼写。

迁移练习

  1. date fromin doubt as to 各写一句话,描述一件你感兴趣的历史文物。
  2. 将以下中文翻译成英文,尽量使用本文句式:"虽然这部古籍的部分内容可能写于唐代,但学者们对其真实年代仍存有疑虑。"
  3. 洛书的幻和是 15。如果一个四阶幻方使用 1–16,它的幻和是多少?用英文写出你的计算过程。
点击展开参考答案
  1. (示例)This ancient bronze vessel is dated from the Shang Dynasty, but archaeologists remain in doubt as to its exact function.
  2. (示例)Although parts of this ancient text may have been written during the Tang Dynasty, scholars are still in doubt as to its genuine age.
  3. Magic constant = n(n² + 1)/2 = 4(16 + 1)/2 = 4 × 17/2 = 34.