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image from Wikipedia with 勾股幂合以成弦幂 vertically on the left, and a square containing right-angled triangles; 朱幂 is written 8 times and 黄幂 (which I may be misreading) is written once in the middle

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

勾股幂合以成弦幂

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?

4 Answers 4

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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

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 , the long side is called , and the side forming a right angle with them is .

is shorter than , and is shorter than .(勾 <= 股股 < 弦

enter image description here


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

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.

  • represents the short right-angled side, like a in the formula.
  • represents the long right-angled side, like b in the formula.
  • 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.
  • 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..
    • means 用来, represents the word to in English.
    • means 变成 or 形成, represents become or form in English.
  • represents the hypotenuse of a right triangle, like c in the formula.
  • as above.

Origin of

about (towel) :

巾,佩巾也。——《说文解字》

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 (power) :

大巾谓之幂。——《小尔雅·广诂》

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

According to the structure of , 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,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 .


Extending this meaning, anything that is square can also be called a . 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.

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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 (power) of the field. Where length and width are multiplied together they are called .)

(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 , then subtract it from the self-multiplier of the , take the square root of the remainder, and the result is .

[3]

自乘之数曰幂。——《几何原本》by 徐光启, 利马窦


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.

and each represent a color, and represents the area of a certain figure.

means 赤色, indicate red.

朱,赤心木,松柏属。——《说文解字》

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 from the value of .

enter image description here


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

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  • 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.
    – LIFeng
    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。"
    – LIFeng
    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
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the character is “冪” (u+51aa)

勾股冪合以成弦冪

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

按弦圖,又可以勾股相乘為朱實二,倍之為朱實四,以勾股之差自相乘為中黃實,加差實,亦成弦實

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

句, or 勾: the shorter right angle side of a right triangle

股: the longer right angle side of a right triangle

弦: the hypotenuse of a right triangle

冪: the square of a number, in mathematics “數學中一數自乘的「次方」”

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

勾股相乘為朱冪二

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

(勾 3 x 股 4) = 朱冪 6 x 2

倍之為朱冪四

double the “勾股相乘” equal to four “朱冪”

2 x (勾 3 x 股 4) = 朱冪 6 x 4

以勾股之差自相乘為中黃實

(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

加差實,亦成弦實

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

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  • 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?
    – Pedroski
    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
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冪in mathematics means 數學上指同一數自乘若干次的乘方。如2自乘四次,就是2的四次冪。。It doesn't seem to apply here.

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.

勾股定理 and 勾股定理 tell how to interpret the graph.

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  • 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
    – Mike M
    Nov 17 at 12:23
  • 1
    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。
    – joehua
    Nov 17 at 16:52
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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。

朱幂 and 黄幂 are colored areas (red and yellow) in the figure. I think the reprint of the book should be in color so that people can understand better.

幂 is the product of two numbers, which also means the area of a square or rectangle.

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