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Note that 冪 is the traditional Chinese character for 幂, and the 幂 mentioned below all refer to 冪.

勾股幂合以成弦幂

variant:

1. 合勾股幂以成弦幂

2. 以勾股幂合成弦幂

According to ancient grammar, there are two variations of this sentence, and I prefer the former.

• 勾 represents the short right-angled side, like a in the formula.
• 股 represents the long right-angled side, like b in the formula.
• 幂 means the product of two numbers multiplied together, here denoting self-multiplication, like the concept of square in modern math.
  • 勾股幂 can be understood as the omitted expression of 勾幂和股幂.
• 合 means 结合, represents combine. It can be simply understood here as the sum of 勾幂 and 股幂.
• 以成 means 用来形成, represents to become/form in English..
  • 以 means 用来, represents the word to in English.
  • 成 means 变成 or 形成, represents become or form in English.
• 弦 represents the hypotenuse of a right triangle, like c in the formula.
• 幂 as above.

Origin of 幂

about 巾（towel） :

巾，佩巾也。——《说文解字》

巾 is a pictographic character. In the oracle bone script, its glyphs resemble a piece of fabric hanging down at both ends, indicating a textile for wiping, covering, wrapping, wearing, etc., much like modern towel.

about 幂（power） :

大巾谓之幂。——《小尔雅·广诂》

幂 is a phonetic word, and its sound side is 冥, indicating that the word and 冥 have similarities in pronunciation. Its shape is 巾, indicating that the word is related to the word 巾.

According to the structure of 幂 , it can be inferred that its original meaning is a towel used to cover things. It is then derived to mean to cover(覆盖，遮掩), and from that to mean area(面积).

about 冖 :

“冖，覆也，从一下垂也。” ——《说文解字》

Cover something with a square piece of cloth and let the four corners hang down to form the shape of an 冖.

Extending this meaning, anything that is square can also be called a 幂. By further extension, the area of a rectangle(矩形)¹ or the product of two numbers²(especially the result of multiplying a number by itself ^[3]) is also called a power. This promotion began with Liu Hui(an ancient mathematician called 刘徽 in China).

"the product of two numbers" means 两数之积 in Chinese. "multiplying a number by itself" means 自乘（一个数与自身相乘） in Chinese.

1

方田术曰：广纵步数相乘得积步。此积谓田幂。凡广纵相乘谓之幂。——《九章算术·方田》

译文 长方形田面积法则：宽与长的步数相乘得积步。（刘徽注：这个积称为田的幂。凡是长宽相乘就称为幂。）

The area rule of a rectangular field: multiply the number of steps of width and length to get the product of steps. (Liu Hui note: This product is called the 幂(power) of the field. Where length and width are multiplied together they are called 幂.)

步 (step) : In ancient times, each of the two feet crossed once called step, now refers to the distance between the two feet when walking.

2

又，勾自乘，以减弦自乘。其余，开方除之，即股。勾、股幂合以成弦幂，令去其一，则余在者皆可得而知之。——《九章算术·勾股》

译文：计算勾的自乘数，然后用弦的自乘数减去它，把余数作开方（根号）运算，结果就是股。

Calculate the self-multiplier of the 勾, then subtract it from the self-multiplier of the 弦, take the square root of the remainder, and the result is 股.

[3]

自乘之数曰幂。——《几何原本》by 徐光启, 利马窦

Now go back to the Pythagorean theorem(毕达哥拉斯定理，勾股定理).

Typically, the mathematical expression we use to describe the Pythagorean Theorem is a^2 + b^2 = c^2.

At the same time, we usually think of the hypotenuse of a right triangle as c, the short right side as a, and the long right side as b.

Therefore, a is shorter than b, and b is shorter than c.(a <= b, b < c)

Description of Pythagorean theorem in ancient Chinese mathematics

短面曰勾，长面曰股，相与结角曰弦。勾短其股，股短其弦。——《九章算术·勾股》

译文： 在直角三角形中，短的边称为勾，长的边称为股，与它们分别形成直角的边为弦。勾的长度比股短，股的长度比弦短。

In a right triangle, the short side is called 勾, the long side is called 股, and the side forming a right angle with them is 弦.

勾 is shorter than 股, and 股 is shorter than 弦.（勾 <= 股，股 < 弦）

What does 朱幂 and 黄幂 mean in this ancient illustration of the Pythagorean Theorem?

about 朱幂 and 黄幂 :

黄 here means yellow.

方 here means square, can be replaced by the word 幂.

As can be seen from the picture:

朱幂 = 1/2 × (勾 × 股) = 1/2 × ab

黄幂 = (勾 - 股) × (勾 - 股) = (b - a)^2

弦幂 = 弦 × 弦 = c^2

The first kind of argument:

Four congruent right triangles form a square(弦幂) with side length c, and in the middle of the figure there is a small square(勾股差幂) with side length b – a.

because:

弦幂 = 4 × 朱幂 + 黄幂

thus:

c^2 = 1/2 × ab × 4 + (b - a)^2

c^2 = 2ab + a^2 + b^2 - 2ab

finally:

c^2 = a^2 + b^2

The second argument:

Then splice four identical congruent right triangles(朱幂) outside the square(弦幂) with side length c, and you will have a square with side length a + b, which can also prove the Pythagorean theorem.

because:

(勾股差幂 + 4 × 黄幂 ) + 4 × 黄幂 = (a + b)^2

弦幂 + 4 × 黄幂 = (a + b)^2

thus:

c^2 + 4 × (1/2 ab) = (a + b)^2

c^2 + 2 × ab = a^2 + b^2 + 2 × ab

finally:

c^2 = a^2 + b^2