#math history

[mathhombre]

\int

Supposedly the first appearance of the integral symbol in Leibniz's notes.

[is-newtonswife]

Recent doodles (and wips)

[niteshade925]

April 14, Xi'an, China, Shaanxi History Museum, Qin and Han Dynasties Branch (Part 3 – Innovations and Philosophies):

(Edit: sorry this post came out so late, I got hit by the truck named life and had to get some rest, and this post in itself took some effort to research. But anyway it's finally up, please enjoy!)

A little background first, because this naming might lead to some confusions.....when you see location adjectives like "eastern", "western", "northern", "southern" added to the front of Zhou dynasty, Han dynasty, Song dynasty, and Jin/晋 dynasty, it just means the location of the capital city has changed. For example Han dynasty had its capital at Chang'an (Xi'an today) in the beginning, but after the very brief but not officially recognized "Xin dynasty" (9 - 23 AD; not officially recognized in traditional Chinese historiography, it's usually seen as a part of Han dynasty), Luoyang became the new capital. Because Chang'an is geographically to the west of Luoyang, the Han dynasty pre-Xin is called Western Han dynasty (202 BC - 8 AD), and the Han dynasty post-Xin is called Eastern Han dynasty (25 - 220 AD). As you can see here, in these cases this sort of adjective is simply used to indicate different time periods in the same dynasty.

Model of a dragonbone water lift/龙骨水车, Eastern Han dynasty. This is mainly used to push water up to higher elevations for the purpose of irrigation:

Model of a water-powered bellows/冶铁水排, Eastern Han dynasty. Just as the name implies, as flowing water pushes the water wheel around, the parts connected to the axle will pull and push on the bellows alternately, delivering more air to the furnace for the purpose of casting iron.

The Nine Chapters on the Mathematical Art/《九章算术》, Fangcheng/方程 chapter. It’s a compilation of the work of many scholars from 10 th century BC until 2 nd century AD, and while the earliest authors are unknown, it has been edited and supplemented by known scholars during Western Han dynasty (also when the final version of this book was compiled), then commented on by scholars during Three Kingdoms period (Kingdom of Wei) and Tang dynasty. The final version contains 246 example problems and solutions that focus on practical applications, for example measuring land, surveying land, construction, trading, and distributing taxes. This focus on practicality is because it has been used as a textbook to train civil servants. Note that during Han dynasty, fangcheng means the method of solving systems of linear equations; today, fangcheng simply means equation. For anyone who wants to know a little more about this book and math in ancient China, here’s an article about it. (link goes to pdf)

Diagram of a circle in a right triangle (called “勾股容圆” in Chinese), from the book Ceyuan Haijing/《测圆海镜》 by Yuan-era mathematician Li Ye/李冶 (his name was originally Li Zhi/李治) in 1248.  Note that Pythagorean Theorem was known by the name Gougu Theorem/勾股定理 in ancient China, where gou/勾 and gu/股 mean the shorter and longer legs of the right triangle respectively, and the hypotenuse is named xian/弦 (unlike what the above linked article suggests, this naming has more to do with the ancient Chinese percussion instrument qing/磬, which is shaped similar to a right triangle).  Gougu Theorem was recorded in the ancient Chinese mathematical work Zhoubi Suanjing/《周髀算经》, and the name Gougu Theorem is still used in China today.

Diagram of the proof for Gougu Theorem in Zhoubi Suanjing. The sentence on the left translates to "gou (shorter leg) squared and gu (longer leg) squared makes up xian (hypotenuse) squared", which is basically the equation a² + b² = c². Note that the character for "squared" here (mi/幂) means "power" today.

This is a diagram of Zhang Heng’s seismoscope, called houfeng didong yi/候风地动仪 (lit. “instrument that measures the winds and the movements of the earth”).  It was invented during Eastern Han dynasty, but no artifact of houfeng didong yi has been discovered yet, this is presumably due to constant wars at the end of Eastern Han dynasty.  All models and diagrams that exist right now are what historians and seismologists think it should look like based on descriptions from Eastern Han dynasty. This diagram is based on the most popular model by Wang Zhenduo that has an inverted column at the center, but this model has been widely criticized for its ability to actually detect earthquakes. A newer model that came out in 2005 with a swinging column pendulum in the center has shown the ability to detect earthquakes, but has yet to demonstrate ability to reliably detect the direction where the waves originate, and is also inconsistent with the descriptions recorded in ancient texts. What houfeng didong yi really looks like and how it really works remains a mystery.

Xin dynasty bronze calipers, the earliest sliding caliper found as of now (not the earliest caliper btw). This diagram is the line drawing of the actual artifact (right).

Ancient Chinese "Jacquard" loom (called 提花机 or simply 花机 in Chinese, lit. "raise pattern machine"), which first appeared no later than 1st century BC. The illustration here is from the Ming-era (1368 - 1644) encyclopedia Tiangong Kaiwu/《天工开物》. Basically it's a giant loom operated by two people, the person below is the weaver, and the person sitting atop is the one who controls which warp threads should be lifted at what time (all already determined at the designing stage before any weaving begins), which creates patterns woven into the fabric. Here is a video that briefly shows how this type of loom works (start from around 1:00). For Hanfu lovers, this is how zhuanghua/妆花 fabric used to be woven, and how traditional silk fabrics like yunjin/云锦 continue to be woven. Because it is so labor intensive, real jacquard silk brocade woven this way are extremely expensive, so the vast majority of zhuanghua hanfu on the market are made from machine woven synthetic materials.

Chinese purple is a synthetic pigment with the chemical formula BaCuSi2O6. There's also a Chinese blue pigment. If anyone is interested in the chemistry of these two compounds, here's a paper on the topic. (link goes to pdf)

A list of common colors used in Qin and Han dynasties and the pigments involved. White pigment comes from chalk, lead compounds, and powdered sea shells; green pigment comes from malachite mineral; blue pigment usually comes from azurite mineral; black comes from pine soot and graphite; red comes from cinnabar; ochre comes from hematite; and yellow comes from realgar and orpiment minerals.

Also here are names of different colors and shades during Han dynasty. It's worth noting that qing/青 can mean green (ex: 青草, "green grass"), blue (ex: 青天, "blue sky"), any shade between green and blue, or even black (ex: 青丝, "black hair") in ancient Chinese depending on the context. Today 青 can mean green, blue, and everything in between.

Western Han-era bronze lamp shaped like a goose holding a fish in its beak. This lamp is interesting as the whole thing is hollow, so the smoke from the fire in the lamp (the fish shaped part) will go up into the neck of the goose, then go down into the body of the goose where there's water to catch the smoke, this way the smoke will not be released to the surrounding environment. There are also other lamps from around the same time designed like this, for example the famous gilt bronze lamp that's shaped like a kneeling person holding a lamp.

Part of a Qin-era (?) clay drainage pipe system:

A list of canals that was dug during Warring States period, Qin dynasty, and pre-Emperor Wu of Han Han dynasty (475 - 141 BC). Their purposes vary from transportation to irrigation. The name of the first canal on the list, Hong Gou/鸿沟, has already become a word in Chinese language, a metaphor for a clear separation that cannot be crossed (ex: 不可逾越的鸿沟, meaning "a gulf that cannot be crossed").

Han-era wooden boat. This boat is special in that its construction has clear inspirations from the ancient Romans, another indication of the amount of information exchange that took place along the Silk Road:

A model that shows how the Great Wall was constructed in Qin dynasty. Laborers would use bamboo to construct a scaffold (bamboo scaffolding is still used in construction today btw, though it's being gradually phased out) so people and materials (stone bricks and dirt) can get up onto the wall. Then the dirt in the middle of the wall would be compressed into rammed earth, called hangtu/夯土. A layer of stone bricks may be added to the outside of the hangtu wall to protect it from the elements. This was also the method of construction for many city walls in ancient China.

A list of the schools of thought that existed during Warring States period, their most influential figures, their scholars, and their most famous works. These include Confucianism (called Ru Jia/儒家 in Chinese; usually the suffix "家" at the end denotes a school of thought, not a religion; the suffix "教" is that one that denotes a religion), Daoism/道家, Legalism (Fa Jia/法家), Mohism/墨家, etc.

The "Five Classics" (五经) in the "Four Books and Five Classics" (四书五经) associated with the Confucian tradition, they are Shijing/《诗经》 (Classic of Poetry), Yijing/《易经》 (also known as I Ching), Shangshu/《尚书》 (Classic of History), Liji/《礼记》 (Book of Rites), and Chunqiu/《春秋》 (Spring and Autumn Annals). The "Four Books" (四书) are Daxue/《大学》 (Great Learning), Zhongyong/《中庸》 (Doctrine of the Mean), Lunyu/《论语》 (Analects), and Mengzi/《孟子》 (known as Mencius).

And finally the souvenir shop! Here's a Chinese chess (xiangqi/象棋) set where the pieces are fashioned like Western chess, in that they actually look like the things they are supposed to represent, compared to traditional Chinese chess pieces where each one is just a round wooden piece with the Chinese character for the piece on top:

A blind box set of small figurines that are supposed to mimic Shang and Zhou era animal-shaped bronze vessels. Fun fact, in Shang dynasty people revered owls, and there was a female general named Fu Hao/妇好 who was buried with an owl-shaped bronze vessel, so that's why this set has three different owls (top left, top right, and middle). I got one of these owls (I love birds so yay!)

And that concludes the museums I visited while in Xi'an!

[foone]

fun fact: You wanna know how you calculated trigonometric functions and logarithms (very handy for complex multiplications!) before calculators?

You had tables!

yeah it's just a book of lots of numbers. these were worked out by hand. It's hundreds of pages. This (along with your slide-rule) was essential to every math student.

This one is a 1959 reprint of a 1943 original

[farouche-landau]

Delighted to share the news that Kurt Hensel was Fanny Mendelssohn Hensel's grandson with someone else who appreciates the work of both! I am very fond of Kurt Hensel's bookplate, which shows his love of music as well as mathematics:

(note the elliptic curve on the paper to the left of the violin!) And yes, German academia was a small world back in the day, and the Mendelssohn family was particularly well-connected: you may already know that Fanny and Felix's younger sister Rebecka married Dirichlet, and their cousin Ottilie married Kummer. And extending the family tradition, one of Kurt Hensel's grandsons, Walter Hayman, also became a mathematician, after leaving Germany for England at age 12 to escape the Nazis. He had a pretty remarkable life (his autobiography is on my to-read list, but it's not available as an e-book, so I will actually have to go to the university library to get it...): among his many accomplishments, he helped his wife Margaret found the British Mathematical Olympiad and send the first British team to the International Math Olympiad. And one of Walter's daughters, Sheila Hayman, became a filmmaker, and directed two films about her family: I still haven't seen Fanny: the Other Mendelssohn, but her earlier film Mendelssohn, the Nazis, and Me covers her family history and the impact of the Nazis in a moving way.

[adragonsfriend]

Mace Windu would've fucking loved graph theory. Legitimately. Math nerd in a modern AU. he's obsessed with finding graphs with higher toughness. he's gonna find a graph of toughness 3 if it kills him.

Alternatively he can be the Václav Chvátal of his world and introduce the concept itself

(this post is about shatterpoints)

[kszug]

So I knew Galois died when he was just 18, but it turns out Abel died when he was just 26? What's up with all these people with immense contributions to group theory dying when they were young????

[miscellaneousmathematics]