Content

Notation of Large Numbers

Shushu Jiyi is the earliest known record of ancient Chinese methods for writing large numbers:

Ancient Calculation Tools

Taiyi Suan (太一算): Taiyi's path involves nine directions. A wooden board is horizontally engraved with nine lines, and a vertical rod has one bead placed on it. Numbers are counted from bottom to top.

Liangyi Suan (兩儀算): A wooden board is horizontally engraved with five lines, and two beads are placed on the vertical line. The upper bead is blue, and the lower bead is yellow. The blue bead counts down from 5 to 1 as it moves downward, while the yellow bead counts up from 1 to 4 as it moves upward.

Sanchen Suan (三才算): A wooden board is horizontally engraved with three lines, representing heaven, earth, and humanity. Three beads are used: a blue one for heaven, a yellow one for earth, and a white one for humanity. The blue bead represents 9 in heaven, 6 in earth, and 3 in humanity. The yellow bead represents 8 in heaven, 5 in earth, and 2 in humanity. The white bead represents 7 in heaven, 4 in earth, and 1 in humanity.

Wuxing Suan (五行算): This method uses the principle of generation and transformation to create endless variations. Zhen Luan's annotation: "The method of the five elements: water is represented by black and has a generating number of one, fire is represented by red and has a generating number of two, wood is represented by blue and has a generating number of three, metal is represented by white and has a generating number of four, earth is represented by yellow and has a generating number of five. For the Wuxing Suan, each color has nine beads that are paired with the corresponding elements to form the calculation positions. For example, for 987654321, white beads paired with yellow represent 900 million, blue beads paired with yellow represent 80 million, red beads paired with yellow represent 700 thousand, black beads paired with yellow represent sixty, one yellow bead represents five million, ten white beads represent forty thousand, one blue bead represents three hundred, one red bead represents twenty, and one black bead represents one."

Bagua Suan (八卦算): Needles point to eight directions, with the ninth position left empty. Zhen Luan's annotation: "For this method, each position is determined by a needle. The number starts from Li (south) as one, then goes to Kun (southwest) as two, Dui (west) as three, Qian (northwest) as four, Kan (north) as five, Gen (northeast) as six, Zhen (east) as seven, and Xun (southeast) as eight. The ninth position is left empty and points to the center, upward towards heaven."

Jiugong Suan (九宮算): This method uses a square with two numbers on each shoulder, six and eight on the feet, three on the left, seven on the right, nine at the top, one at the bottom, and five in the center. The positions are set according to the five elements.

: For each position, a calculation rod of five inches is used, with five notches on it. Each notch represents five units when close to one end, and one unit for each subsequent notch moving away from the end. The positions are determined by the spaces between the joints of the fingers: the first joint of the index finger represents the ones place, the second joint represents the tens place, and so on.

: Each position is represented by a single "Liao" character. The three curves in the Liao character represent different numbers: the bottom curve's end represents one internally and nine externally, the first upper curve represents two internally and eight externally, the second upper curve represents three internally and seven externally, the third upper curve represents four internally and six externally, and the top of the Liao character represents five.

: Each position requires a bead of one of five colors. The head of each bead is yellow, representing the generating number, while other parts represent different numbers according to the elements: water (black) generates one and completes six, fire (red) generates two and completes seven, wood (blue) generates three and completes eight, metal (white) generates four and completes nine. For example, for 98,765,432,1, a white bead with the head pointing north represents 900 million, a blue bead with the head pointing west represents 80 million, a red bead with the head pointing north represents seven hundred thousand, a black bead with the head pointing west represents sixty, and so on. The direction of the head determines whether it represents a generating or completing number.

: This method requires two beads: one plain and one notched. The notched bead has four sides with numbers 1 to 4, while the plain bead serves as the head, representing five units. The notched side represents individual units.

Area Measurement

Indeterminate Analysis

Buddhism and Taoism

Zhen Luan was influenced by the "Three Teachings Confluence" philosophy of Southern Dynasties Taoist Tao Hongjing. The phrase "doubling repeatedly" in the book is from the Avatamsaka Sutra; terms like "kṣaṇa" and "brahma" are also derived from Buddhist scriptures; "seven dusts of water make one dust on a rabbit's hair, seven dusts on a rabbit's hair make one dust on a sheep's hair..." is from the Lankavatara Sutra. The phrase "Huangdi established the law, with ten levels of numbers..." is quoted from the Taoist figure Zheng Xuan, and the term "Dayan" comes from the I Ching.

Editions

Southern Song dynasty, 1212 CE: The only extant edition is preserved at Peking University Library. It was reprinted by Cultural Relics Press in 1980.

Ming dynasty, Wanli period: The "Secret Book Collection" (秘冊匯函) edition edited by Hu Zhenheng.

"Siku Quanshu" (四庫全書) edition with annotations by Dai Zhen.

Dai Zhen and Kong Jihan's "Micro Wave Pavilion" (微波榭) edition of the "Ten Mathematical Classics."

Qian Baozong's annotated version of the "Ten Mathematical Classics," published by Zhonghua Book Company based on Dai Zhen and Kong Jihan's "Micro Wave Pavilion" edition.

Qian Baozhong's annotated Shushu Jiyi , included in "The Complete Works of Li Yan and Qian Baozong in the History of Science," Volume 4, pages 407-416.

Gou Shuchun and Liu Dun's annotated edition published by Liaoning Education Press.

References

• 數術記遺 (Chinese)

Source: ''Shushu Jiyi (數術記遺)'', also known as ''Sushu Jiyi (術數記遺)'', is a single-volume work written by Xu Yue of the Eastern Han dynasty. It was annotated by Zhen Luan, the governor of Hanzhong Commandery (who previously served as an Inspector) during the Northern Zhou dynasty. The work was listed as a required textbook for the Ming Suan (明算) examination in the Tang dynasty, which contributed to its preservation. ==Content== ==Notation of Large Numbers== Shushu Jiyi is the earliest known record of ancient Chinese methods for writing large numbers: ==Ancient Calculation Tools== ''Taiyi Suan (太一算)'': Taiyi's path involves nine directions. A wooden board is horizontally engraved with nine lines, and a vertical rod has one bead placed on it. Numbers are counted from bottom to top. ''Liangyi Suan (兩儀算)'': A wooden board is horizontally engraved with five lines, and two beads are placed on the vertical line. The upper bead is blue, and the lower bead is yellow. The blue bead counts down from 5 to 1 as it moves downward, while the yellow bead counts up from 1 to 4 as it moves upward. ''Sanchen Suan (三才算)'': A wooden board is horizontally engraved with three lines, representing heaven, earth, and humanity. Three beads are used: a blue one for heaven, a yellow one for earth, and a white one for humanity. The blue bead represents 9 in heaven, 6 in earth, and 3 in humanity. The yellow bead represents 8 in heaven, 5 in earth, and 2 in humanity. The white bead represents 7 in heaven, 4 in earth, and 1 in humanity. ''Wuxing Suan (五行算)'': This method uses the principle of generation and transformation to create endless variations. Zhen Luan's annotation: "The method of the five elements: water is represented by black and has a generating number of one, fire is represented by red and has a generating number of two, wood is represented by blue and has a generating number of three, metal is represented by white and has a generating number of four, earth is represented by yellow and has a generating number of five. For the Wuxing Suan, each color has nine beads that are paired with the corresponding elements to form the calculation positions. For example, for 987654321, white beads paired with yellow represent 900 million, blue beads paired with yellow represent 80 million, red beads paired with yellow represent 700 thousand, black beads paired with yellow represent sixty, one yellow bead represents five million, ten white beads represent forty thousand, one blue bead represents three hundred, one red bead represents twenty, and one black bead represents one." ''Bagua Suan (八卦算)'': Needles point to eight directions, with the ninth position left empty. Zhen Luan's annotation: "For this method, each position is determined by a needle. The number starts from Li (south) as one, then goes to Kun (southwest) as two, Dui (west) as three, Qian (northwest) as four, Kan (north) as five, Gen (northeast) as six, Zhen (east) as seven, and Xun (southeast) as eight. The ninth position is left empty and points to the center, upward towards heaven." ''Jiugong Suan (九宮算)'': This method uses a square with two numbers on each shoulder, six and eight on the feet, three on the left, seven on the right, nine at the top, one at the bottom, and five in the center. The positions are set according to the five elements. Tao Hongjing. The phrase "doubling repeatedly" in the book is from the Avatamsaka Sutra; terms like "kṣaṇa" and "brahma" are also derived from Buddhist scriptures; "seven dusts of water make one dust on a rabbit's hair, seven dusts on a rabbit's hair make one dust on a sheep's hair..." is from the Lankavatara Sutra. The phrase "Huangdi established the law, with ten levels of numbers..." is quoted from the Taoist figure Zheng Xuan, and the term "Dayan" comes from the I Ching. ==Editions== Southern Song dynasty, 1212 CE: The only extant edition is preserved at Peking University Library. It was reprinted by Cultural Relics Press in 1980. Ming dynasty, Wanli period: The "Secret Book Collection" (秘冊匯函) edition edited by Hu Zhenheng. "Siku Quanshu" (四庫全書) edition with annotations by Dai Zhen. Dai Zhen and Kong Jihan's "Micro Wave Pavilion" (微波榭) edition of the "Ten Mathematical Classics." Qian Baozong's annotated version of the "Ten Mathematical Classics," published by Zhonghua Book Company based on Dai Zhen and Kong Jihan's "Micro Wave Pavilion" edition. Qian Baozhong's annotated Shushu Jiyi , included in "The Complete Works of Li Yan and Qian Baozong in the History of Science," Volume 4, pages 407-416. Gou Shuchun and Liu Dun's annotated edition published by Liaoning Education Press. ==References== * 數術記遺 (Chinese) {{AI-translation}}