3-4-5 三角形问题The 3-4-5 Triangle Problem

常用词汇

同义词与近义词

常用短语与固定搭配

• lay out：布置，规划，铺设。原文用法："ancient Egyptian surveyors laid out right angles by constructing 3–4–5 triangles..."
• to the effect that：大意是，内容为。原文用法："no documentary evidence to the effect that the Egyptians were aware of..."
• be aware of：知道，意识到。原文用法："the Egyptians were aware of even a particular case of the Pythagorean theorem"
• care to do something：愿意做某事（常用于礼貌邀请）。原文用法："The reader may care to try to solve this problem."
• based on：基于，以……为基础。原文用法："A particularly simple solution can be effected based on Figure 3..."
• make up：组成，构成。原文用法："...along with a unit square, making up a square of 25 square units of area."
• it follows that：由此可见，从而得出。原文用法："...and it follows that a 3–4–5 triangle is a right triangle."

常用句式

• There are reports that...：有报告称……（表达未经完全证实的信息）。原文例句："There are reports that ancient Egyptian surveyors laid out right angles..."
• Since there is no evidence to the effect that..., ...：既然没有……内容的证据，那么……（证据缺失 → 谨慎结论）。原文例句："Since there is no documentary evidence to the effect that the Egyptians were aware of..., the following purely academic problem arises"
• Show, without using ..., that...：证明/说明……，但不使用……（数学问题中的限制条件表述）。原文例句："Show, without using the Pythagorean theorem, its converse, or any of its consequences, that the 3–4–5 triangle is a right triangle."
• This means that..., and it follows that...：这意味着……，从而得出……（推理链中的连续推导）。原文例句："This means that the hypotenuse...is 5 units long, and it follows that a 3–4–5 triangle is a right triangle."

长难句解析

"Since there is no documentary evidence to the effect that the Egyptians were aware of even a particular case of the Pythagorean theorem, the following purely academic problem arises: Show, without using the Pythagorean theorem, its converse, or any of its consequences, that the 3-4-5 triangle is a right triangle."

• 主干：problem arises（问题出现了）
• 修饰与从句：

- "Since there is no documentary evidence to the effect that..." —— 原因状语从句 - "to the effect that the Egyptians were aware of..." —— 修饰 evidence 的不定式短语，内含同位语从句说明 evidence 的内容 - "the following purely academic problem arises:" —— 主句 - "Show, without using..., that..." —— 冒号后的问题陈述（Show 的宾语从句 "that the 3-4-5 triangle is a right triangle"） - "without using the Pythagorean theorem, its converse, or any of its consequences" —— 插入的限制性状语

• 逻辑关系：原因（Since 无证据）→ 结果（问题 arises：需在不使用 XXX 的前提下证明 YYY），典型的"限制条件下的证明问题"表述。

"A particularly simple solution can be effected based on Figure 3, which shows four right triangles with legs of 3 and 4 units, along with a unit square, making up a square of 25 square units of area."

• 主干：A particularly simple solution can be effected（一个特别简单的解法可以被实现/给出）
• 修饰与从句：

- "based on Figure 3," —— 方式状语 - "which shows four right triangles with legs of 3 and 4 units, along with a unit square," —— 非限制性定语从句，修饰 Figure 3 - "making up a square of 25 square units of area." —— 现在分词短语作结果状语，说明图中各元素共同组成的大图形

• 逻辑关系：方法（based on Figure 3）→ 图的内容（which shows...）→ 结果（making up a square of 25...）→ 结论（solution can be effected）

语言使用特点

• 本文属于数学问题陈述（mathematical problem statement） 文体，特点是：条件限定精确（"without using..."）、术语使用规范（converse, consequence, hypotenuse, legs, generalize, arbitrary）、逻辑衔接紧密（Since → arises → Show → This means → it follows）。
• 使用了典型的数学指令动词：Show（证明/说明）、effected（实现/完成）。这些动词在数学英语中有固定的搭配模式。
• 书名使用斜体并附英文译名（Chou-pei Suan-king (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven)），符合学术引用规范。
• "may care to try" 是一种委婉的表达方式，相当于"读者不妨试试"，体现了作者对读者的尊重和鼓励态度。

迁移练习

1. 用 there is no evidence to the effect that 写一句话，描述一个历史上尚未证实的事件。
2. 将以下中文翻译成英文，尽量使用本文句式："由于没有文献记载表明古希腊人知道无理数的一般定义，以下问题自然出现了：证明 √2 不能表示为两个整数的比值。"
3. 用英文描述图 3 的构造过程，至少使用 legs、hypotenuse、unit square、make up 这四个词语。