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

The above is an illustration of the Pythagorean Theorem from the work 周髀算经. On the left it says:

Which I think is a 8-character statement of the Pythagorean Theorem, as follows:

• 勾 ("adjacent" in modern terminology)
• 股 ("opposite" in modern terminology)
• 幂 ("square", but now it means "exponent")
• 合 ("sum", but is now written 和)
• 以成 (something like "gives" or "becomes")
• 弦 ("hypotenuse" in modern terminology)
• 幂 ("square", as above)

Maybe I'm not 100% accurate here, but that makes logical sense to me. But I have no idea what 朱幂 nor 黄幂 (?) means and why it is written on this diagram.

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

The Pythagorean theorem(translation is `毕达哥拉斯定理`) in ancient Chinese mathematics is called `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

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`

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` 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） :

`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

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

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]

### Now back to the question.

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

#### about `朱幂` and `黄幂` :

`朱幂` and `黄幂` here are names representing variables in the illustration.

`朱` and `黄` each represent a color, and `幂` represents the area of a certain figure.

`朱` means `赤色`, indicate red.

`黄` here means yellow.

The eight `朱幂` in the illustration refer to the eight right triangles.

The `黄幂` refers to the middle square.

The `弦幂` I mentioned is a variable representing the large square around the perimeter.

The value of `朱幂` is the value of the area of the right triangle.

The value of `黄幂` is the value of the area of the small square in the middle.

The value of `弦幂` is the sum of four `朱幂` and one `黄幂`.

In the picture I provided, the new variable `勾股差幂` refers to `黄幂`.

`勾股差幂` can better express the meaning of this small square.

Its value is the square of the result obtained by subtracting the value of `勾` from the value of `股`.

As can be seen from the illustration provided by the questioner:

`朱幂` = `1/2 × (勾 × 股)` = `1/2 × ab`

`黄幂` = `(股 - 勾) × (股 - 勾)` = `(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`

• the 國語辭典 stated that “幂” is a variant of “冪” 😼 dict.revised.moe.edu.tw/dictView.jsp?ID=30960&word=幂 Nov 16 at 15:11
• Where is 青 coming from? It’s not mentioned anywhere else except in your definition that it means ‘yellow’. Was it supposed to say 黄? Nov 16 at 18:26
• @水巷孑蠻 You are correct, "冪" is the traditional character for "幂". In ancient Chinese, "冪" should be used. The use of "幂" in the title led me to overlook this. Nov 17 at 8:54
• @JanusBahsJacquet: I'm sorry that my mistake misled you. The word "青" comes from the book "九章算术·勾股" in the sentence "勾自乘为朱方，股自乘为青方。", from what I've read. Although the illustration shows "黄", the first word mentioned in the original text is "青". "青" is often used together with "黄" in ancient Chinese, such as "青黄不接", so it may also mean yellow. It also means black ("青丝"), green ("青松") and blue ("青天"). In addition, the "黄幂" in the illustration comes from the sentence at the back of the original text, "一丈自乘为朱幂四、黄幂一。半差自乘，又倍之，为黄幂四分之二，减实，半其余，有朱幂二、黄幂四分之一。" Nov 17 at 8:56
• @LIFeng And it’s composed of 生 and 丹, which means red. Conclusion: the Ancient Chinese were clearly colour blind! Nov 17 at 11:14

the character is “冪” (u+51aa)

the picture is generally called “勾股方圓圖”, made by 趙爽; it’s function is to prove the 勾股定理 (pythagorean theorem) by geometry 😸

most likely, there’s accompanied text

at this moment, i’m not sure it’s “實”, or “冪”

## how to comprehend the picture:

it’s on a 7 x 7 grid, so

“黃冪” is the area of the centre square, which is 1 x 1 = 1

“朱冪” is the area of a right triangle, which is 3 x 4 / 2 = 6

and, there’re 8 such right triangles in the picture

the shorter side x the longer side equal to two “朱冪”

(勾 3 x 股 4) = 朱冪 6 x 2

double the “勾股相乘” equal to four “朱冪”

2 x (勾 3 x 股 4) = 朱冪 6 x 4

(the longer side subtract the shorter side) x (the longer side subtract the shorter side) equal to the area of the centre square

(4 - 3) x (4 - 3) = 1

now, looking at the square with bold side (the rhombus?)

it’s composed by 4 right triangles + the centre square, so, it’s area is:

4 x 朱冪 + 黃冪

4 x (3 x 4 / 2) + 1 = 25 = 5 x 5

that, a triangle with 3 & 4 as its side, the hypotenuse would be 5

have fun :)

edited

• Pythagoras was extended by Einstein to incorporate time: ds2 = dx2 + dy2 + dz2 -dt2 which very simply shows: at the speed of light, there is no time, ds = 0! 周髀没有时间吗? Nov 16 at 16:37
• “弦: the of a right triangle” – the what of a right triangle? Nov 16 at 18:20
• @JanusBahsJacquet, ooops, my apologies, sahib 😸 it means the hypotenuse of a right triangle 😸 Nov 17 at 0:04

Although in the picture, it reads 朱冪 and 黃冪，if you read the text, it actually says 朱實 and 黃實。

The entry 17 of 實 in 漢典 says "古代数学名词。指被乘数或被除数。与“法”相对。如以3除6或乘6，则6为实，3为法。《九章算術•序》：“以景差為法，表高乘表閒為實，如法而一。”" So, 實 can mean either multiplicand or dividend. However, that annotation doesn't seem to apply here either.

I think it can conveniently be explained as area, red area and yellow area.

• while the maths are cool, i can't help but imagine this is closer to what the OP is asking :) I wonder if the original had color, or if the colors were stock phrases for labels Nov 17 at 12:23
• Here is the original text of 《周髀算經》。You see it says on the side 朱實六黃實一。Here is a scanned page from《周髀算經注》. It does say 朱實，此層實朱色。On the next page, you see 此層實青色，此層實黃色。 Nov 17 at 16:52

I guess, 勾股幂合, 以成弦幂. 勾的幂与股的幂合并，就成为弦的幂。