内容概述
传统一致认为，毕达哥拉斯独立发现了以他名字命名的直角三角形定理。不过早在汉谟拉比时代，巴比伦人已知道该定理；一般性的第一个证明则可能由毕达哥拉斯给出。后人推测其证明可能是“分割式证明”：将两个边长为 a+b 的大正方形分别分割成六块与五块，通过“等量减等量”得到斜边上正方形面积等于两直角边上正方形面积之和。要说明第二次分割中央部分确为边长 c 的正方形，还需用到直角三角形内角和等于两直角的性质。普罗克洛斯将一般情形下的该定理归功于毕达哥拉斯学派。
中文正文
45° 毕达哥拉斯证明他的定理
直角三角形斜边的平方等于两个直角边的平方之和，这个以毕达哥拉斯的名字命名的定理，一直被公认为他的独立发现。我们在故事 17 里看到，早在 1000 多年前，汉谟拉比时代的巴比伦人就已经知道这个定理了，不过它的第一个一般性的证明，则很可能是毕达哥拉斯提出的。至于毕达哥拉斯的证明，历来有许多猜想，一般认为它可能属于如下图所示的那种分割式的证明方法。令 a，b，c 为给定的直角三角形的直角边和斜边，考虑图中的两个正方形，边长都是 a+b。第一个正方形被分割成 6 块：两个以直角边为底的正方形，四个全等于原来的直角三角形。第二个正方形被分割为 5 块：一个以斜边为底的正方形和四个全等于原来的直角三角形。从相等的量中减去相等的量，我们就看到，斜边正方形的面积等于两个直角边正方形的面积之和。
为了证明第二个分割中央的那块东西确实是边长为 c 的正方形，我们需要借助一个事实：直角三角形三个角之和等于两个直角。关于这个定理的一般情形，希腊评论家普罗克洛斯（Proclus）却把它归功于毕达哥拉斯。因为这个定理的证明需要知道一些平行线的性质，而人们也相信毕达哥拉斯发展过那个理论。
18 在《周髀算经》开篇，商高答周公问时说，大禹治水时代就有了“故折矩以为勾，广三，股修四，径隅五”的“勾股定理”。如果那对话是真的，应该在公元前 1100 年左右。
英文正文
45° Pythagoras's proof of his theorem. Tradition is unanimous in ascribing to Pythagoras the independent discovery of the theorem on the right triangle that now universally bears his name, that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs. We have seen, in Item 17°, that this theorem was known to the Babylonians of Hammurabi's time, more than a thousand years earlier, but the first general proof of the theorem may well have been given by Pythagoras. There has been much conjecture as to the proof Pythagoras might have offered, and it is generally felt that it probably was a dissection type of proof* like the following, illustrated in Figure 5. Let a, b, c denote the legs and hypotenuse of the given right triangle, and consider the two squares in the figure, each having a + b as side. The first square is dissected into six pieces, namely the two squares on the legs and four right triangles congruent to the given triangle. The second square is dissected into five pieces, namely the square on the hypotenuse and again four right triangles congruent to the given triangle. By subtracting equals from equals, it now follows that the square on the hypotenuse is equal to the sum of the squares on the legs.
To prove that the central piece of the second dissection is actually a square of side c we need to employ the fact that the sum of the angles of a right triangle is equal to two right angles. But the Greek commentator Proclus attributes this theorem for the general triangle to the Pythagoreans. Since a proof of this theorem requires, in turn, a knowledge of some properties of parallels, the early Pythagoreans are also credited with the development of that theory.
See, however, Daniel Shanks, Solved and Unsolved Problems in Number Theory*, Vol. 1 (Washington, D.C.: Spartan Books, 1962), pp. 124, 125.
英语学习拓展
常用词汇
unanimous音标/juˈnænɪməs/
一致同意的。
原文用法：Tradition is unanimous in ascribing...
ascribe音标/əˈskraɪb/
把……归因于。
原文用法：同上
independent音标/ˌɪndɪˈpendənt/
独立的。
原文用法：the independent discovery of the theorem
universally音标/ˌjuːnɪˈvɜːrsəli/
普遍地。
原文用法：now universally bears his name
bear音标/ber/
带有；承载。
原文用法：同上
hypotenuse音标/haɪˈpɑːtənuːs/
斜边。
原文用法：the square on the hypotenuse of a right triangle
leg音标/leɡ/
直角边。
原文用法：the sum of the squares on the two legs
conjecture音标/kənˈdʒektʃər/
猜想。
原文用法：There has been much conjecture as to the proof
dissection音标/dɪˈsekʃn/
分割。
原文用法：a dissection type of proof
denote音标/dɪˈnoʊt/
表示；代表。
原文用法：Let a, b, c denote the legs and hypotenuse
congruent音标/ˈkɑːŋɡruənt/
全等的。
原文用法：four right triangles congruent to the given triangle
subtract音标/səbˈtrækt/
减去。
原文用法：By subtracting equals from equals
employ音标/ɪmˈplɔɪ/
使用；运用。
原文用法：we need to employ the fact
commentator音标/ˈkɑːmənteɪtər/
评论家。
原文用法：the Greek commentator Proclus
attribute音标/əˈtrɪbjuːt/
把……归功于。
原文用法：Proclus attributes this theorem ... to the Pythagoreans
parallel音标/ˈpærəlel/
平行线。
原文用法：some properties of parallels
同义词与近义词
对应核心词汇：ascribe、attribute
对应核心词汇：ascribe。
辨析：ascribe 常指主观归因；attribute 强调正式认定来源；credit 指给予荣誉或认可。
对应核心词汇：denote
对应核心词汇：denote。
辨析：denote 指精确地表示；represent 泛指代表；stand for 侧重象征意义。
对应核心词汇：congruent
对应核心词汇：congruent。
辨析：congruent 为几何术语，指形状大小完全相同；identical 强调完全相同；equal 指数值或地位相等。
常用短语与固定搭配
- be unanimous in doing：一致做某事。原文用法：Tradition is unanimous in ascribing to Pythagoras...
- ascribe to：把……归于。原文用法：同上
- bear one's name：以某人的名字命名。原文用法：now universally bears his name
- be known to：为……所知。原文用法：this theorem was known to the Babylonians...
- dissection type of proof：分割式证明。原文用法：a dissection type of proof
- be congruent to：与……全等。原文用法：four right triangles congruent to the given triangle
- subtract equals from equals：等量减等量。原文用法：By subtracting equals from equals
- attribute ... to ...：把……归功于……。原文用法：Proclus attributes this theorem ... to the Pythagoreans
- be credited with：被认为有……成就。原文用法：the early Pythagoreans are also credited with the development of that theory
常用句式
- Pattern: Tradition is unanimous in ascribing ... to ...：传统一致认为……属于……。原文例句：Tradition is unanimous in ascribing to Pythagoras the independent discovery of the theorem...
- Pattern: Let ... denote ...：令……表示……。原文例句：Let a, b, c denote the legs and hypotenuse of the given right triangle...
- Pattern: By doing ..., it follows that ...：通过做……，可以得出……。原文例句：By subtracting equals from equals, it now follows that ...
长难句解析
Tradition is unanimous in ascribing to Pythagoras the independent discovery of the theorem on the right triangle that now universally bears his name, that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs.
- 主干：Tradition is unanimous in ascribing to Pythagoras the independent discovery of the theorem
- 修饰与从句：
- on the right triangle 修饰 theorem； - that now universally bears his name 为定语从句，修饰 theorem； - that the square on the hypotenuse ... is equal to the sum of the squares on the two legs 为同位语从句，具体说明 theorem 的内容。
- 逻辑关系：主句说明传统观点（将独立发现归于毕达哥拉斯），随后用两个 that 从句分别说明定理的命名和具体表述。
语言使用特点
- 使用大量数学术语（hypotenuse、congruent、dissection），体现学术文本特征；
- 用 Tradition is unanimous、it is generally felt 等表达说明历史共识；
- 证明叙述采用命令式（Let ... denote、consider）和因果推理（By subtracting ... it follows that）。
迁移练习
- 用 bear one's name 写一个句子，说明某条定理或定律以谁命名。
- 解释 subtract equals from equals 在证明中的含义。
- 翻译：令 a、b、c 分别表示给定直角三角形的直角边和斜边。
点击展开参考答案
- （示例）The Pythagorean theorem bears the name of the ancient Greek mathematician Pythagoras.
- （示例）It means if you take equal quantities away from equal quantities, the remaining quantities are equal. In the proof, removing the four congruent triangles from the two equal large squares leaves the squares on the sides of the triangle.
- Let a, b, c denote the legs and hypotenuse of the given right triangle.