Proposal to encode Ancient Chinese Mathematical Symbols Kushim Jiang ( 姜兆勤 ) (kushim_jiang@outlook.com) July 12, 2017 1. Introduction of the new block - Ancient Chinese Mathematical Symbols. Because Chinese mathematics emerged independently, when writing mathematical books, Chinese people created many special symbols. Especially in Ming and Qing Dynasty, a lot of western mathematics books were translated verbally and introduced into China by western missionaries. In order to adapt to the vertical writing of Chinese books, the Chinese native scholars modified the western mathematical symbols slightly and then a set of new symbols were made and used into books and papers. These changes are mainly reflected in the following aspects: Mathematical Symbols. Most of them were just rotated but some of them were reformed. Name of unknowns. The Latin letters and the Greek letters will be translated into the names of Chinese era, twentyeight lunar mansions or the sixty-four hexagrams. Equations. The simple equations would be written vertically, and the complex equations would be written horizontally, sometimes the column could be so long to contain the equations and sometimes an equation could cross several columns. If there are mathematical symbols which had been reformed and used in China before China drew up standards, we will put them into this new block. Considering that there are also many unexplored books, we set the new block from U+1EF00 to U+1EF3F. 1.2. Introduction of specific blocks. (1) Combining Diacritical Marks. Like point Aʹ, point Aʺ in western symbol system, when expressing some similar and different geometric points, ancient Chinese mathematician would use some left-falling strokes ( 撇 ). One to three left-falling strokes were found in Qiuyishu Tongjie ( 求一術通解 ). (2) Combining Diacritical Marks for Symbols. The UCS contains a number of combining diacritical marks for symbols. For example, can be represented as 〤 𝍩 〩〩, where (U+20E5) is used for indicating a negative number. There are also some marks in Xuesuan Bitan ( 學算筆談 ), Siyuan Yujian ( 四元玉鑑 ) and Zhusuan ( 珠算 ). In the books, is used for deleting or updating the number during the mathematical operation, and is used for marking the redundant characters in the process of copying and printing, and is used as! in math, which means factorial. For example, both 五 and 5! represent 5 4 3 2 1. (3) Enclosed Ideograms. The UCS contains a number of enclosed ideograms in BMP and SMP. In ancient Chinese math books, the ideograms enclosed by circles are also found in Pingsanjiao Bianjiao Huqiushu ( 平三角邊角互求術 ), Weiji Suyuan ( 微積溯源 ), Qiuyishu Tongjie ( 求一術通解 ), Duishu Xiangjie ( 對數詳解 ), Daishushu ( 代數術 ) and Zhusuan ( 珠算 ). And the ideograms enclosed by squares are also found in Yuanlü Kaozhen Tujie ( 圓率攷眞圖解 ), Daishu Beizhi ( 代數備旨 ), Subu Yancao ( 粟布演草 ), Duishu Xiangjie ( 對數詳解 ) and Siyuan Yujian Xicao ( 四元玉鑑細草 ). In general, most of the characters are used for marking the formula so as to quote and discuss them in later words. (4) Ancient Chinese Mathematical Symbols. A brief introduction is given here. a), and. means plus (+), means minus (-), and means plus or munus (±). They were adopted in
Daiweiji Shiji ( 代微積拾級 ). For saving the space, the and usually closed to the characters narrowly. For example a + b would be represented as 甲 乙. So the usage of them obviously differ from UP TACK ( ), DOWN TACK ( ) and PERPENDICULAR SIGN ( ) and COUNTING RODS (𝍥, 𝍮). b) and. is the variant of, and is the variant of. They were adopted in Xingxue Beizhi ( 形學備旨 ) and Daishu Beizhi ( 代數備旨 ). And the usage of differ from NONFORKING ( ). c). means find positive difference. For example, 5 3 = 3 5 = 2. d) and. They are the variants of { and }. e). means et cetera ( ). The explanatory notes of Duishu Xiangjie ( 對數詳解 ) enumerated this symbol particularly, for it hasn t been used before. f) and. is deformed from the radical of 微, which means differential (d). is deformed from the radical of 積, which means integral ( ). And the usage of them as operators is different from them as radicals, so they should be separated. 2. Character Data. 2.1. Character position and name. Combining Diacritical Marks Extended (U+1AB0 - U+1AFF) 1AC0 1AC1 1AC2 COMBINING CHINESE PRIME COMBINING CHINESE DOUBLE PRIME COMBINING CHINESE TRIPLE PRIME Combining Diacritical Marks for Symbols (U+20D0 - U+20FF) 20F1 20F2 20F3 COMBINING LONG SOLIDUS OVERLAY COMBINING ENCLOSING OCTAGON COMBINING CHINESE FACTORIAL SYMBOL Ancient Chinese Mathematical Symbols (U+1EF00 - U+1EF3F) 1EF00 CHINESE PLUS SIGN 1EF01 CHINESE MINUS SIGN 1EF02 1EF03 1EF04 1EF05 CHINESE PLUS-OR-MINUS SIGN CHINESE VARIANT PLUS SIGN CHINESE VARIANT MINUS SIGN CHINESE POSITIVE DIFFERENCE SIGN 1EF06 CHINESE LEFT CURLY BRACKET
1EF07 CHINESE RIGHT CURLY BRACKET 1EF08 1EF09 1EF0A CHINESE ELLIPSIS CHINESE DIFFERENTIAL SIGN CHINESE INTEGRAL SIGN U+5341,U+034F,U+4E00,U+034F,U+25EF U+5341,U+034F,U+4E8C,U+034F,U+25EF U+5341,U+034F,U+4E09,U+034F,U+25EF U+5341,U+034F,U+56DB,U+034F,U+25EF U+5341,U+034F,U+4E94,U+034F,U+25EF U+5341,U+034F,U+516D,U+034F,U+25EF U+5341,U+034F,U+4E03,U+034F,U+25EF U+5341,U+034F,U+516B,U+034F,U+25EF U+5341,U+034F,U+4E5D,U+034F,U+25EF U+7532,U+034F,U+25EF U+4E59,U+034F,U+25EF U+4E19,U+034F,U+25EF U+4E01,U+034F,U+25EF U+620A,U+034F,U+25EF U+5DF1,U+034F,U+25EF U+5E9A,U+034F,U+25EF U+8F9B,U+034F,U+25EF U+5477,U+034F,U+25EF U+20B99,U+034F,U+25EF
U+20C33,U+034F,U+25EF U+53EE,U+034F,U+25EF U+3595,U+034F,U+25EF U+5B50,U+034F,U+25EF U+4E11,U+034F,U+25EF U+5BC5,U+034F,U+25EF U+536F,U+034F,U+25EF U+8FB0,U+034F,U+25EF U+5DF3,U+034F,U+25EF U+5348,U+034F,U+25EF U+672A,U+034F,U+25EF U+7533,U+034F,U+25EF U+9149,U+034F,U+25EF U+4EA5,U+034F,U+25EF U+89D2,U+034F,U+25EF U+4EA2,U+034F,U+25EF U+6C10,U+034F,U+25EF U+623F,U+034F,U+25EF U+4E8C,U+034F,U+7533,U+034F,U+25EF U+4E8C,U+034F,U+9149,U+034F,U+25EF U+4E8C,U+034F,U+4EA5,U+034F,U+25EF U+4E8C,U+034F,U+5929,U+034F,U+25EF U+7532,U+034F,U+4E8C,U+034F,U+25EF
U+4E59,U+034F,U+4E8C,U+034F,U+25EF U+4E19,U+034F,U+4E8C,U+034F,U+25EF U+4E01,U+034F,U+4E8C,U+034F,U+25EF U+620A,U+034F,U+4E8C,U+034F,U+25EF U+5DF1,U+034F,U+4E8C,U+034F,U+25EF U+5E9A,U+034F,U+4E8C,U+034F,U+25EF U+8F9B,U+034F,U+4E8C,U+034F,U+25EF U+5B50,U+034F,U+4E8C,U+034F,U+25EF U+4E11,U+034F,U+4E8C,U+034F,U+25EF U+5BC5,U+034F,U+4E8C,U+034F,U+25EF U+536F,U+034F,U+4E8C,U+034F,U+25EF U+58F9,U+034F,U+25EF U+8CB3,U+034F,U+25EF U+53C1,U+034F,U+25EF U+5343,U+034F,U+25EF U+5929,U+034F,U+25EF U+5730,U+034F,U+25EF U+4EBA,U+034F,U+25EF U+8853,U+034F,U+25EF U+53C8,U+034F,U+25EF U+3007,U+034F,U+2B1C U+56DB,U+034F,U+2B1C U+4E94,U+034F,U+2B1C
U+516D,U+034F,U+2B1C U+4E03,U+034F,U+2B1C U+516B,U+034F,U+2B1C U+4E5D,U+034F,U+2B1C U+5341,U+034F,U+2B1C U+5341,U+4E8C,U+034F,U+2B1C U+5341,U+56DB,U+034F,U+2B1C U+5341,U+516D,U+034F,U+2B1C U+5341,U+4E03,U+034F,U+2B1C U+4E7E,U+034F,U+2B1C U+5426,U+034F,U+2B1C U+554F,U+034F,U+2B1C U+5C65,U+034F,U+2B1C U+5C6F,U+034F,U+2B1C U+5E2B,U+034F,U+2B1C U+6B65,U+034F,U+2B1C U+6BD4,U+034F,U+2B1C U+6CF0,U+034F,U+2B1C U+8A1F,U+034F,U+2B1C U+4749,U+034F,U+2B1C U+96A8,U+034F,U+2B1C U+9700,U+034F,U+2B1C It is important to point out that this kind of enclosed ideographic has fixed form and certain system, so they cannot be added or reduced casually.
The reason why is different from it in the evidence figures is that, there were mistakes in books in the process of transcribing. Because the serial characters were marked in the Twelve Earthly Branches order and 巳 in the Ten Heavenly Stems cannot be in the serial for no reason, the correct character should be 己. 2.2. Character properties. 1AC0;COMBINING CHINESE PRIME;Mn;230;NSM;;;;;N;;;;; 1AC1;COMBINING CHINESE DOUBLE PRIME;Mn;230;NSM;;;;;N;;;;; 1AC2;COMBINING CHIENSE TRIPLE PRIME;Mn;230;NSM;;;;;N;;;;; 20F1;COMBINING LONG SOLIDUS OVERLAY;Mn;1;NSM;;;;;N;;;;; 20F2;COMBINING ENCLOSING OCTAGON;Me;0;NSM;;;;;N;;;;; 20F3;COMBINING CHINESE FACTORIAL SYMBOL;Mn;230;NSM;;;;;N;;;;; 1EF00;CHINESE PLUS SIGN;Sm;0;L;;;;;N;;;;; 1EF01;CHINESE MINUS SIGN;Sm;0;L;;;;;N;;;;; 1EF02;CHINESE PLUS-OR-MINUS SIGN;Sm;0;L;;;;;N;;;;; 1EF03;CHINESE VARIANT PLUS SIGN;Sm;0;L;;;;;N;;;;; 1EF04;CHINESE VARIANT MINUS SIGN;Sm;0;L;;;;;N;;;;; 1EF05;CHINESE POSITIVE DIFFERENCE SIGN;Sm;0;L;;;;;N;;;;; 1EF06;CHINESE LEFT CURLY BRACKET;Sm;0;L;;;;;N;;;;; 1EF07;CHINESE RIGHT CURLY BRACKET;Sm;0;L;;;;;N;;;;; 1EF08;CHINESE ELLIPSIS;Sm;0;L;;;;;N;;;;; 1EF09;CHINESE DIFFERENTIAL SIGN;Sm;0;L;;;;;N;;;;; 1EF0A;CHINESE INTEGRAL SIGN;Sm;0;L;;;;;N;;;;; 3. References. 刘鐸辑. 古今算學叢書 ( 据微波榭本等石印 ). 上海 : 算學書局, 光緒二十四年 (1898). 劉永錫, ( 美 ) Calvin Wilson Mateer. 形學備旨 ( 十卷 ). 重校本. 葉耀元. 形學補編 ( 一卷 ). 原稿本. 吳嘉善. 平三角術 ( 一卷 ). 白芙堂叢書本. ( 英 ) John Fryer. 三角須知 ( 一卷 ). 原刊本. 左潛. 綴術釋明附刊誤 ( 二卷 ). 白芙堂叢書本. 曾紀鴻, 左潛, 黃宗憲. 圓率考真圖解附刊誤 ( 一卷 ). 白芙堂叢書本. 黃宗憲, 左潛. 求一術通解 ( 二卷 ). 白芙堂叢書本. 戴煦. 對數簡法附刊誤 ( 二卷 ) 續對數簡法附刊誤 ( 一卷 ). 小萬卷樓叢書本. 曾紀鴻, 丁取忠. 對數詳解附刊誤 ( 五卷 ). 白芙堂叢書本. 李善蘭, ( 附卷 ) 王季同. 測圓海鏡圖表附九容公式 ( 一卷 ). 原稿本. 李冶, ( 校 ) 李銳. 益古演段 ( 三卷 ). 白芙堂叢書本. 朱世傑, ( 細草 ) 羅士琳. 四元玉鑑細草 ( 二十四卷 ). 觀我生室彙稿本. 華蘅芳, ( 英 ) John Fryer. 代數術附刊誤 ( 二十五卷 ). 江南製造局本. [ 第一百十四款, 補劉彝程 沈善蒸 崔朝慶算式 ] 鄒立文, 生福維. 代數備旨 ( 六卷 ). 重校本. 李善蘭, ( 英 ) Alexander Wylie, ( 注 ) 湯金鑄, 李鳳苞, 華蘅芳. 代微積拾級注 ( 十八卷 ). 墨海刊本. 附三家注原稿本. 華蘅芳, ( 英 ) John Fryer. 微積溯源 ( 六卷 ). 江南製造局本. 秦九韶, ( 附卷 ) 宋景昌, ( 校 ) 鄒安鬯. 數書九章十八卷附札記 ( 四卷 ). 宜稼堂叢書本校注原稿本. ( 第一卷 ) 丁取忠, 左潛, 曾紀鴻, 吳嘉善, 李善蘭, ( 第二卷 ) 鄒伯奇, 丁取忠, 左潛, ( 補卷 ) 丁取忠. 粟布演草 ( 二卷 ) 補 ( 一卷 ). 白芙堂叢書本. 方中通. 珠算 ( 一卷 ). 隨衍室本. 徐建寅, ( 英 ) John Fryer. 運規約指 ( 三卷 ). 江南製造局本. 華蘅芳. 學算筆談 ( 十二卷 ). 原刻本. (There are also 75 books collected in the series but not put to use for examples) 4. Acknowledgement. This proposal is improved with great help and advice of Eduardo Marin and Andrew West. And the glyphs of the
symbols are modified from Symbola and Sursong. 5. Figures. Figure 1. Example from Huang Zongxian s ( 黃宗憲 ) book Qiuyishu Tongjie ( 求一術通解 ) vol.2 p.7 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing, and in text.
Figure 2. Example from Fang Zhongtong s ( 方中通 ) book Zhusuan ( 珠算 ) p.16 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing in text. Figure 3. Example from Zhu Shijie s ( 朱世傑 ) book Siyuan Yujian ( 四元玉鑑 ) vol.1.2 p.11 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing in text.
Figure 4. Example from Hua Hengfang s ( 華蘅芳 ) book Xuesuan Bitan ( 學算筆談 ) vol.8 p.4 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing in text. Figure 5. Example from Wu Jiashan s ( 吳嘉善 ) book Pingsanjiao Bianjiao Huqiushu ( 平三角邊角互求術 ) p.1 and p.2 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,,, and in text.
Figure 6. Example from Hua Hengfang s ( 華蘅芳 ) book Weiji Suyuan ( 微積溯源 ) vol.6 p.19, p.44 and p.45 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,, and in text. Figure 7. Example from Hua Hengfang s ( 華蘅芳 ) book Daishushu ( 代數術 ) vol.7 p.7 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing and in text.
Figure 8. Example from Huang Zongxian s ( 黃宗憲 ) book Qiuyishu Tongjie ( 求一術通解 ) vol.2 p.7 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,, and in text.
Figure 9. Example from Huang Zongxian s ( 黃宗憲 ) book Qiuyishu Tongjie ( 求一術通解 ) vol.2 p.12 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,,, and in text. Figure 10. Example from Zeng Jihong s ( 曾紀鴻 ) book Duishu Xiangjie ( 對數詳解 ) vol.4 p.3 and p.4 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing, and in text.
Figure 11. Example from Hua Hengfang s ( 華蘅芳 ) book Daishushu ( 代數術 ) vol.8 p.10; vol.24 p.13, p.15, p.19 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,,,, and in text. Figure 12. Example from Hua Hengfang s ( 華蘅芳 ) book Daishushu ( 代數術 ) vol.24 p.22, p.26 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing and in text.
Figure 13. Example from Hua Hengfang s ( 華蘅芳 ) book Daishushu ( 代數術 ) vol.24 p.27, p.29, p.31, p.32, p.33, p.34 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,,,,,, and in text. Figure 14. Example from Hua Hengfang s ( 華蘅芳 ) book Daishushu ( 代數術 ) vol.24 p.35, p.37, p.39, p.40 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,, and in text.
Figure 15. Example from Hua Hengfang s ( 華蘅芳 ) book Daishushu ( 代數術 ) vol.25 p.6, p.7, p.8, p.17, p.19 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,,,, and in text. Figure 16. Example from Hua Hengfang s ( 華蘅芳 ) book Daishushu ( 代數術 ) vol.25 p.22, p.25 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,, and in text.
Figure 17. Example from Fang Zhongtong s ( 方中通 ) book Zhusuan ( 珠算 ) p.5 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing, and in text. Figure 18. Example from Zeng Jihong s ( 曾紀鴻 ) book Yuanlü Kaozhen Tujie ( 圓率攷眞圖解 ) p.13 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,, and in text.
Figure 19. Example from Zou Liwen s ( 鄒立文 ) book Daishu Beizhi ( 代數備旨 ) vol.3 p.13 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing in text. Figure 20. Example from Zeng Jihong s ( 曾紀鴻 ) book Subu Yancao ( 粟布演草 ) vol.1 p.3 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing, and in text.
Figure 21. Example from Zeng Jihong s ( 曾紀鴻 ) book Duishu Xiangjie ( 對數詳解 ) vol.2 p.7, p.8 and p.10 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,,, and in text. Figure 22. Example from Zeng Jihong s ( 曾紀鴻 ) book Duishu Xiangjie ( 對數詳解 ) vol.3 p.1, p.2 and p.3 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,, and in text.
Figure 23. Example from Zeng Jihong s ( 曾紀鴻 ) book Duishu Xiangjie ( 對數詳解 ) vol.4 p.2, p.3, p.4 and p.6 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,,,, and in text. Figure 24. Example from Alexander Wylie and Li Shanlan s ( 李善蘭 ) book Daiweiji Shiji Zhu ( 代微積拾級注 ) vol.1 p.10 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing,, and in text.
Figure 25. Example from Zou Liwen s ( 鄒立文 ) book Xingxue Beizhi ( 形學備旨 ) vol.1 p.11 and Daishu Beizhi ( 代數備旨 ) vol.1 p.19 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing and in text. Figure 26. Example from Hua Hengfang s ( 華蘅芳 ) book Xuesuan Bitan ( 學算筆談 ) vol.8 p.1 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing in text.
Figure 27. Example from Zeng Jihong s ( 曾紀鴻 ) book Duishu Xiangjie ( 對數詳解 ) vol.1 p.2 and p.3 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing, and in text. Figure 28. Example from Hua Hengfang s ( 華蘅芳 ) book Weiji Suyuan ( 微積溯源 ) vol.6 p.36 (Shanghai: Arithmetic Publishing House ( 算学书局 ), 1898), showing and in text.