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Title: Zhou Bi Suan Jing
Author: Unknown

Release date: May 1, 2004 [eBook #12408]
Most recently updated: October 28, 2024

Language: Chinese

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Zhou Bi Suan Jing

100 B.C.

Volume 1 of Zhou Bi Suan Jing
In ancient times, Duke Zhou asked Shang Gao, “I have heard that scholars are skilled in mathematics.”

May I ask how in ancient times the Zhou calendar was established?
The heavens cannot be climbed step by step, nor can the earth be measured with exact dimensions.
Where do numbers originate from?
Shang Gao said, “The method of numbers comes from circles and squares. Circles come from squares, and squares come from rectangles. Rectangles arise from 9 multiplied by 9, which equals 81. Thus, a rectangle is folded to form a segment with a width of three, a height of four, and a diagonal of five. After its outer sides are squared, half of a rectangle is taken, and it is circled around to form a circle, resulting in the numbers 3, 4, and 5. Two such rectangles combined have a total length of twenty-five; this is called the ‘accumulated rectangle.’ This is how Yu governed the world—through these numbers.”
Duke of Zhou said, “Great are the words concerning numbers!”
May I ask about the uses of rectangles?
Shang Gao said, “Use a horizontal rectangle to straighten ropes, an inverted rectangle to measure heights, a flipped rectangle to determine depths, and a laid-out rectangle to gauge distances. Circle a rectangle to form a circle, and combine rectangles to form a square. Squares relate to the earth, while circles relate to the heavens. The heavens are round and the earth is square. Square numbers serve as standards; from squares, circles are derived. A cone is used to represent the heavens—the heavens are blue and black, while the earth is yellow and red. The numbers of the heavens correspond to the cone, with blue and black representing the exterior and red and yellow representing the interior, symbolizing the positions of heaven and earth. Therefore, those who understand the earth possess wisdom, while those who understand the heavens are sage. Wisdom arises from segments, and segments arise from rectangles. As for rectangles in relation to numbers, they are used to shape all things according to one’s intent.”
Duke of Zhou said, “Well said.”

Zhou Bi Suan Jing, Volume 1, Part 2
In ancient times, Rong Fang asked Chen Zi, “I have heard of the master’s teachings—how one can know the height and size of the sun, the range of its light, the distance it travels in a day, the distances at various ranges, what can be seen from afar, the extremities of the earth, the positions of the stars, and the vastness of heaven and earth. Can all these truly be known through the master’s teachings? Is this true?”
Chen Zi replied, “Yes.”
Rong Fang said, “Although I do not fully understand, I hope the master will kindly explain it to me. Can someone like me be taught these teachings?”
Chen Zi said, “Yes. All these fall within the scope of arithmetic. With sufficient knowledge of arithmetic, one can understand these things. If one truly ponders over them carefully…”
Rong Fang went back and thought about it for several days but could not figure it out. He then went to see Chen Zi again and said, “I have tried to think about it but still cannot understand. I dare to ask for your guidance.” Chen Zi said, “Your thinking is not yet thorough enough. This is also a technique related to seeing distant things from heights, yet you cannot grasp it. It shows that your understanding of numbers is not comprehensive enough. There are limits to your wisdom and insight.”
The teachings of the Way are concise in expression but have broad applications; they require clarity of understanding. Those who can understand one aspect and apply it to countless situations truly understand the Way. What you are learning now—arithmetic—is a use of wisdom, yet it still presents difficulties. This shows that your range of understanding is limited. The reason why the teachings of the Way are difficult to grasp is that even after learning them, one may lack breadth of knowledge; even if one has breadth, one may lack practice.

It has already been learned; the problem lies in not being able to understand it.
Therefore, those with similar skills learn from each other, and those working on the same tasks observe one another.
This is how the wise and the foolish are distinguished, and how the excellent and the inferior are separated.
Thus, one can categorize things to bring like things together. This is the essence of the wise’s mastery through diligent practice and wisdom.
Those who study the same subject but cannot achieve mastery lack wisdom and fail to practice diligently.
Hence, their calculations are not precise. How could I hide this truth? I thought about it thoroughly again. Rong Fang thought about it once more, but couldn’t figure it out after several days. He then went to see Master Chen and said, “I’ve thought about it intensively, but there are limits to my wisdom and understanding. I hope you’ll explain it to me.”
Master Chen said, “Sit down again; I’ll tell you.” So Rong Fang sat down and asked for Master Chen’s explanation. He said, “At the summer solstice, it is 16,000 li south; at the winter solstice, it is 135,000 li south. A pole is set up at noon to measure the shadow. This is one of the principles of the heavens’ laws. The Zhou Bi circle is eight feet long; on the day of the summer solstice, the shadow length is 1 foot 6 inches. ‘Bi’ refers to the thigh, and ‘zheng gui’ refers to the shadow length. One thousand li due south, the shadow length is 1 foot 5 inches; one thousand li due north, it is 1 foot 7 inches. As the days progress, the southern shadow grows longer. When the shadow length reaches 6 feet, take a bamboo tube with a diameter of 1 inch and a length of 8 feet, place it over the sun, and observe the shadow. The tube covers the sun, and the sun’s image appears in the hole of the tube. From this, we find that for every 80 inches, the diameter is 1 inch. Therefore, using ‘gui’ as the numerator and ‘bi’ as the denominator, from the point below the sun to the sun itself is 60,000 li, and at that point there is no shadow. From there upward to the sun, it is 80,000 li. Applying this ratio, every 80 li corresponds to a diameter of 1 li; every 100,000 li corresponds to a diameter of 1,250 li. Hence, the diameter of the sundial shadow is 1,250 li. If we want to find the date when the sun reaches an oblique position, take the distance below the sun as the numerator and the distance above the sun as the denominator. Multiply each by itself, add them together, take the square root, and divide to get the date when the sun is at an oblique position. The distance from the side of the ‘bi’ line to the sun is 100,000 li. According to the method, since the Zhou Bi circle is 8 feet long, a change of 1 inch corresponds to 1,000 li. Thus, the extremes represent the vastness of the heavens.”

The current vertical pole is eight feet tall, allowing one to see as far as possible. Its circumference is thirteen inches. From this, it can be calculated that the distance from the north of Zhou to the southernmost point is 130,300 li.

Rong Fang asked, “What is ‘Zhou Bi’?” Chen Zi replied, “In ancient times, the emperor ruled over Zhou. These measurements were taken from Zhou, hence the name ‘Zhou Bi’.” “Bi” refers to a vertical pole.

On the day of the summer solstice, the sun is 16,000 li south; on the day of the winter solstice, it is 135,000 li south. At noon, there is no shadow. From this, it can be deduced that the distance from the south to the noon point on the summer solstice is 119,000 li. The same applies when moving north to midnight. The total straight-line distance is 238,000 li. This is the distance along the path on the day of the summer solstice. Its circumference is 714,000 li.

From the noon point on the summer solstice to the noon point on the winter solstice, it is 119,000 li. The same holds true when moving north to the southernmost point. Thus, from the southernmost point to the noon point on the winter solstice, it is 238,000 li; from the northernmost point to midnight, it is also 238,000 li. The total straight-line distance is 476,000 li. This is the distance along the path on the day of the winter solstice. Its circumference is 1,428,000 li.

From the noon point on the vernal equinox to the southernmost point, it is 178,500 li. From the southernmost point to midnight in the north, it is also 178,500 li. The total straight-line distance is 357,000 li. The circumference is 1,710,000 li.

Therefore, it is said that the moon’s path always follows the constellations, and the sun’s path also aligns with them. From the south to the noon point on the summer solstice, to the north to midnight on the winter solstice, from the south to the noon point on the winter solstice, to the north to midnight on the summer solstice, the straight-line distance is still 357,000 li, with a circumference of 1,710,000 li.

From the night of the vernal equinox to the night of the autumnal equinox, there is always sunlight at the southernmost point. From the night of the autumnal equinox to the night of the vernal equinox, there is no sunlight at the southernmost point.

Thus, during the nights of the vernal and autumnal equinoxes, the area illuminated by sunlight reaches the poles, marking the boundary between yin and yang. The winter and summer solstices are the points where the sun’s path reaches its extremes, determining the longest and shortest days. The vernal and autumnal equinoxes represent the balance between yin and yang and the patterns of day and night. Day corresponds to yang, while night corresponds to yin. From the vernal equinox to the autumnal equinox, it is the time of day; from the autumnal equinox to the vernal equinox, it is the time of night.

Therefore, at the noon points of the vernal and autumnal equinoxes, the light reaches the northernmost point, while at midnight, the light reaches the southernmost point. This is the time of day and night. Hence, it is said that sunlight illuminates areas in all directions, each spanning 167,000 li.

What people can see in distance, near or far, should be like what is illuminated by sunlight.
From the center of the sphere, one can see 64,000 li to the north past the pole.
To the south, on the day of winter solstice, it’s 32,000 li; on the day of summer solstice, it’s 48,000 li beyond the midday light.
To the south, what people can see is 16,000 li.
To the north from the center, it’s 151,000 li; to the north past the pole, it’s 48,000 li.
On the night of winter solstice, there’s sunlight at midnight; to the south, it’s 7,000 li beyond what human eyes can see.
Beyond the pole, it’s 71,000 li.
On the day of summer solstice, from midday to midnight, there’s sunlight for 96,000 li—this extends all the way to the pole.
On the day of winter solstice, from midday to midnight, there’s sunlight for 142,000 li; beyond the pole, it’s 71,000 li.
On the day of summer solstice, when looking due east or west, the sun reaches a distance of 59,598.5 li from the center.
On the day of winter solstice, the sun is not visible in the due east or west directions.
By calculation, the sun reaches a distance of 214,557.5 li from the center.
All these figures relate to the expansion and contraction of the sun’s path.
At winter and summer solstices, by observing the laws and listening to the sound of bells, one can determine the differences and the extent of sunlight.
The diameter at the four poles is 810,000 li, while the circumference is 2,430,000 li.
From the center to the point where sunlight reaches to the south is 320,000 li; to the north, it’s 580,000 li.
East and west each measure 391,163.5 li.
The center is 103,000 li south of the equator; thus, the east-west distance is 26,632 li more than the diameter at the equator.
To the north of the center, it’s 580,000 li; on the day of winter solstice, it’s 135,000 li. The path length on that day is 476,000 li. The total circumference is 1,428,000 li. At the four poles, the distance from the center east and west each measures 391,163.5 li plus a little extra.
This is the method of determining circles and squares.

Volume 3 of Zhou Bi Suan Jing
To create this diagram, use a zhang as a chi, a chi as a cun, and a cun as a fen. One fen equals one thousand li. Originally, a square measuring eight chi and one cun was used; now a square measuring four chi and five fen is used, with one fen equaling two thousand li.
Lü Shi said that within the boundaries of the four seas, the distance from east to west is 28,000 li, and from north to south is 26,000 li.
For the circle representing the motion of the sun and moon, there are seven equal segments and six gaps, corresponding to the six months. Each month consists of 182 days and 8/5 parts of a day. Thus, the summer solstice occurs within the inner segment of Dongjing, while the winter solstice occurs outside the outer segment of Qianniu. The segments alternate until the end of winter. Hence, there are 365 days and 1/4 part of a day in a year. Within a year, there is one inner extreme and one outer extreme, totaling 30 days and 7/16 parts of a day. Each month also has one outer extreme and one inner extreme.
Therefore, the distance between two consecutive segments is 19,833 li and 1/3 part of a li, which equals 100 steps. To find the diameter of the next segment, double the diameter of the previous inner segment, then multiply by 2 again to increase it further. The diameter of the first segment is 238,000 li, with a circumference of 714,000 li. This is divided into 365 degrees and 1/4 part of a degree. Each degree equals 1,954 li, 247 steps, and 933/1,461 parts of a step.
The diameter of the second segment is 277,666 li and 200 steps, with a circumference of 833,000 li. Divided into degrees, each degree equals 2,280 li, 188 steps, and 1,332/1,461 parts of a step.
The diameter of the third segment is 317,333 li and 100 steps, with a circumference of 952,000 li. Divided into degrees, each degree equals 2,660 li, 130 steps, and 270/1,461 parts of a step.
The diameter of the fourth segment is 357,000 li, with a circumference of 1,710,000 li. Divided into degrees, each degree equals 2,932 li, 71 steps, and 669/1,461 parts of a step.
The diameter of the fifth segment is 396,666 li and 200 steps, with a circumference of 1,190,000 li. Divided into degrees, each degree equals 3,258 li, 12 steps, and 168/1,461 parts of a step.
The diameter of the sixth segment is 436,333 li and 100 steps, with a circumference of 1,309,000 li. Divided into degrees, each degree equals 3,583 li, 254 steps, and 6/1,461 parts of a step.

The diameter at the next seven parallels is 476,000 li, and the circumference is 1,428,000 li. It is divided into degrees; each degree equals 3,991 li, 195 steps, and 45/1,461 of a step. Next, at the northernmost point reached by the sun on the winter solstice, it is 167,000 li beyond the Northern Parallel. The diameter there is 810,000 li, and the circumference is 2,430,000 li. It is divided into 365 degrees and 1/4 of a degree; each degree equals 6,652 li, 293 steps, and 327/1,461 of a step. Those who have explored beyond the Northern Parallel do not know this. Those who do know it either doubt whether it can be known or doubt its difficulty in being known. It is said that the highest sages know this without learning it. Therefore, on the winter solstice, the gnomon is 13 feet and 5 inches long; on the summer solstice, it is 6 feet long. The gnomon is longer on the winter solstice and shorter on the summer solstice. A change of one inch in the gnomon results in a difference of 1,000 li. Thus, on the days of the winter and summer solstices, the distance traveled north and south is 119,000 li. The diameter at the four poles is 810,000 li, and the circumference is 2,413,000 li. It is divided into degrees; each degree equals 6,652 li, 293 steps, and 327/1,461 of a step. The distance between these degrees is as follows: the daily travel north and south is 651 li, 182 steps, and 798/1,461 of a step. The method states: take 119,000 li as the numerator and half a year, 182 days plus 5/8 of a day as the denominator, and divide. This yields 952,000 as the numerator and 1,461 as the denominator. Divide the numerator by the denominator to get one li. If the result is less than one unit of the denominator, multiply by three; if it equals one hundred steps, multiply by ten; if it equals ten steps, multiply by ten; if it equals one step, multiply by the value given by the denominator.

Volume 2, Chapter 1 of the Zhou Bi Suan Jing
Regarding the movement of the sun and moon and the paths of the four poles: at the lowest pole, the land is 60,000 li high above where people live, with water pouring down from all sides.

At the center of heaven, it is also four hundred thousand li high and six hundred thousand li wide in all directions. Thus, the area illuminated by sunlight extends eighty-one hundred thousand li, covering a total of two million four hundred thirty thousand li. Therefore, when the sun is at its farthest point in the north, it is midday in the north while it is midnight in the south. When the sun is at the far east, it is midday there while it is midnight in the west. When the sun is at the far south, it is midday there while it is midnight in the north. When the sun is at the far west, it is midday there while it is midnight in the east. These four directions represent the four poles of heaven and earth and their harmonious balance. Day and night alternate, along with the four seasons. Yet the extremes of yin and yang, as well as those of winter and summer, are all the same. Heaven resembles a canopy, and earth resembles a platform. Heaven is eighty thousand li above earth. On the day of the winter solstice, although the sun is at its farthest point, it still appears twenty thousand li above the earth’s surface. Thus, the sun influences the moon, causing its light to appear, resulting in a bright moon. The stars then arrange themselves in their proper positions. From the autumn equinox to the winter solstice, the subtle energies of the three lights travel great distances to reach their destinations. This is the natural nature of yin and yang in heaven and earth. To determine the pole of the north star and its movement around the four poles, one should observe at midnight on the summer solstice—the farthest point the north star reaches to the south. At midnight on the winter solstice, observe its farthest point to the north. At the time when the sun moves toward the west on the winter solstice, observe its farthest point to the west. At the time when the sun moves toward the east, observe its farthest point to the east. These are the four movements of the north star. The true pole of the north star lies at the center of the celestial sphere, directly in the north, at the heart of heaven. At the time when the sun moves toward the west on the winter solstice, set up an eight-foot tall pole and tie a rope to it, then look toward the large star in the center of the north sky. Use the rope to measure the distance from the pole to the ground. By doing this again at the time when the sun moves toward the east the next day, measure the distance between the ends of the rope; it will be two feet and three inches. Thus, the distance between the eastern and western poles is twenty-three thousand li, with the ends being directly east and west of each other.

Fold it in half. Use a finger to indicate, pointing straight north and south.
To determine the times for east, west, south, and north, all are measured using water clocks. The string is tied to the marker, with the marked point being three inches away from it. Thus, the center of the sky is 103,000 li away from the Earth’s center.
To know the times corresponding to the north and south poles: at midnight on the winter solstice, the northernmost point reaches its extreme; it is 11,500 li beyond the center of the sky. At the summer solstice, the southernmost point reaches its extreme, being less than 11,500 li from the center of the sky. All these are determined by tying a string to the marker and observing it. At the north pole, the marked point is 1 foot and 4.5 inches above the ground, so it is 114,500 li away from the Earth’s center, 11,500 li beyond the center of the sky. At the south pole, the marked point is 9 feet and 1.5 inches above the ground, so it is 91,500 li away from the Earth’s center, less than 11,500 li from the center of the sky. This is the method for determining whether the four poles are beyond or within the center of the sky for east, west, south, and north.
The Earth is 103,000 li away from the poles. The Sun is 167,000 li away from humans. At the summer solstice, it is 16,000 li away from the Earth’s center. The distance the Sun travels on that day is 238,000 li, while the Earth’s circumference is 714,000 li. On the vernal and autumnal equinoxes, the distance the Sun travels is 357,000 li, making the Earth’s circumference 1,710,000 li. On the winter solstice, the distance the Sun travels is 436,000 li, giving an Earth’s circumference of 1,428,000 li. The light from the Sun reaches the four poles over a distance of 810,000 li, while the distance around the Earth is 2,430,000 li. From the Earth’s center, it is 302,000 li to the south.
The diameter of the celestial sphere is 23,000 li, while the Earth’s diameter is 69,000 li. This is because yang is dominant and yin is subdued, so no living things can arise there.
The method involves establishing a reference point and fixing it. When the Sun first rises, set up a marker and note the shadow length; do the same when the Sun sets. If the two ends of the shadow are directly opposite each other, it indicates true east-west direction. Folding it in half and using a finger to indicate points to the marker shows true north-south direction. No living things grow below the poles. How is this known? On the winter solstice, it is 119,000 li from the summer solstice, and all living things die. On the summer solstice, it is 119,000 li from the north pole, thus proving that no living things grow below the poles. To the left and right of the north pole, there is ice that does not melt in summer. On the vernal and autumnal equinoxes, the Sun is at the celestial equator. After the vernal equinox, the Sun moves north by 59,500 li until the summer solstice. After the autumnal equinox, it moves south by 59,500 li until the winter solstice. The celestial equator is 75,500 li away from the Earth’s center. To the left and right of the celestial equator, there are plants that do not die in winter and grow in summer. Here, yang is prominent and yin is weak, so all living things survive, and crops can be harvested twice a year. Around the north pole, some plants grow in the morning and are ready to harvest by evening.
The twenty-eight constellations were established using the methods of measuring the celestial sphere.
The method involves doubling the distance due south, then fixing it using the reference point.

That is, the diameter on flat ground is twenty-one. On Saturday, it’s thirteen steps. Make it level and straight with water.
Then the diameter is one hundred twenty-one feet, seven inches, and five fractions. Multiply by three to get three hundred sixty-five feet and one fraction of a foot.
This corresponds to one part in three hundred sixty-five degrees and one fraction of a degree of the celestial circle. Determine the fractions precisely, without any error.
With the degrees determined, the meridians and parallels can be established accurately. One quarter of this equals ninety-one degrees and five and a half fractions of a degree each.
Thus, the circle becomes precise and accurate.
Then set up a pillar at the center to indicate north and south. Tie a rope to its top and align it with the center of the Altair star.
Next, observe which star arrives first. Again, use the rope from the pillar to align it with the star that arrives first.
Then use an armillary sphere to align it with the center of Altair and determine how many degrees west of the pillar it is.
The distance measured by the armillary sphere in feet represents the degree.
Since the armillary sphere is eight feet above, Altair is eight degrees away.
For the subsequent stars, repeat this process to cover all twenty-eight constellations, and thus everything will be determined.
To establish the degrees around the circle, use the degrees indicated by the armillary sphere for each star that arrives first. Use the ropes from the armillary sphere to align them with the center, thereby ensuring accuracy.
The time when the sun sets is also determined using these methods.
To know when the sun rises or sets, observe at midnight when Dongjing is at its highest point and Altair first reaches the position corresponding to Zi. Dongjing rises thirty degrees, sixteen and a half fractions of a degree west of the pillar and reaches the position corresponding to Wei. At that time, Altair should also be at the position corresponding to Chou.
Thus, heaven and earth are in harmony.
Then place the twenty-eight constellations along the circle. Once placed, establish the center of the circle again and set up the pillar.
On the days of winter and summer solstices, observe the moment the sun begins to rise. Place an armillary sphere at that position and observe the shadow cast by the pillar.
When the shadow is aligned correctly, that indicates the degree of the constellation where the sun rises. The same method is used to determine when the sun sets.

Volume 2 of Zhou Bi Suan Jing
Qian Niu: 115 degrees, 1,695 li, 21 steps, 1,461 fractions from the North Pole; 819 of these.
The method states: Set the outer balance at 238,000 li from the North Pole pivot. Divide by 11,500 li. The remainder is 226,500 li, which is taken as the actual value. Use 1,954 li, 247 steps, 1,461 fractions per degree of the inner balance as the divisor. Dividing the actual value by this divisor gives one degree. If the result is less than the divisor, calculate the corresponding number of li and steps by rounding; every 300 gives one unit of the actual value. Use 1,461 fractions as the divisor to get one li. If the result is still less than the divisor, multiply by 3; then divide by the divisor to get 100 steps. If it’s still insufficient, multiply by 10 again to get one step. For any remaining amount, use the divisor to determine the value.

Lou and Jiao: 91 degrees, 610 li, 264 steps, 1,461 fractions from the North Pole; 1,296 of these.
The method states: Set the middle balance at 178,500 li from the North Pole pivot, taking this as the actual value. Use the value per degree of the inner balance as the divisor. Dividing the actual value by this divisor gives one degree. If the result is less than the divisor, calculate the corresponding number of li and steps; for any remaining amount, use the divisor to determine the value.

Dong Jing: 66 degrees, 1,481 li, 155 steps, 1,461 fractions from the North Pole; 1,245 of these.
The method states: Set the inner balance at 119,000 li from the North Pole pivot, add 11,500 li, resulting in 135,000 li as the actual value. Use the value per degree of the inner balance as the divisor. Dividing the actual value by this divisor gives one degree. If the result is less than the divisor, calculate the corresponding number of li and steps; for any remaining amount, use the divisor to determine the value.

In total, there are eight solar terms and twenty-four qi. Each qi increases or decreases by 9 inches, 9 fractions, and 1/6 fraction. At the winter solstice, the shadow length is 13 feet 5 inches; at the summer solstice, it is 1 foot 6 inches. Determine the increase or decrease in inches for each subsequent solar term.

On the Winter Solstice, the shadow length is 13 feet and 5 inches.
On Minor Cold, it is 12 feet and 5 inches, with a minor fraction of 5.
On Major Cold, it is 11 feet and 5 inches 1 fraction, with a minor fraction of 4.
At the Beginning of Spring, it is 5 feet 2 fractions, with a minor fraction of 3.
During Rain Water, it is 9 feet 5 inches 3 fractions, with a minor fraction of 2.
At the Start of Insect Activity, it is 8 feet 5 inches 4 fractions, with a minor fraction of 1.
At the Equinox of Spring, it is 7 feet 5 inches 5 fractions.
During Clear and Bright Days, it is 6 feet 5 inches 5 fractions, with a minor fraction of 5.
At Grain Rain, it is 5 feet 5 inches 6 fractions, with a minor fraction of 4.
At the Beginning of Summer, it is 4 feet 5 inches 7 fractions, with a minor fraction of 3.
At Lesser Fullness, it is 3 feet 5 inches 8 fractions, with a minor fraction of 2.
At Grain in Ear, it is 2 feet 5 inches 9 fractions, with a minor fraction of 1.
On the Summer Solstice, it is 1 foot 6 inches.
During Minor Heat, it is 2 feet 5 inches 9 fractions, with no minor fraction.
At Major Heat, it is 3 feet 5 inches 8 fractions, with a minor fraction of 2.
At the Beginning of Autumn, it is 4 feet 5 inches 7 fractions, with a minor fraction of 3.
At End of Heat, it is 5 feet 5 inches 6 fractions, with a minor fraction of 4.
At White Dew, it is 6 feet 5 inches 5 fractions, with a minor fraction of 5.
At the Equinox of Autumn, it is 7 feet 5 inches 5 fractions, with a minor fraction of 1.
At Cold Dew, it is 8 feet 5 inches 4 fractions, with a minor fraction of 1.
At Frost’s Descent, it is 9 feet 5 inches 3 fractions, with a minor fraction of 2.
At the Beginning of Winter, it is 5 feet 2 fractions, with a minor fraction of 3.
During Light Snow, it is 11 feet 5 inches 1 fraction, with a minor fraction of 4.
At Heavy Snow, it is 12 feet 5 inches, with a minor fraction of 5.
In total, there are eight solar terms and twenty-four climatic periods.
The variation in shadow length is 9 feet 9 inches 6 fractions 1/1.
The Winter Solstice and Summer Solstice mark the beginning of these variations.
According to the method: measure the shadow length on the Winter Solstice, subtract it from that on the Summer Solstice; the remainder gives the actual value. Use 12 as the divisor. If the result is 1 inch, multiply it by 10 and divide by 12 to get the fraction.

Those dissatisfied with the law shall be ordered according to the law.
Thirteen degrees, nineteen minutes, and seven parts after the moon.
The method is: take 235 for the chapter month, divide by 19 for the chapter year, add one degree per day, resulting in thirteen degrees, nineteen minutes, and seven parts. This is the amount advanced on the first day of the month, which is also the degrees and minutes of the subsequent days.
For a minor year, since the month is insufficient, discard 354 degrees, 7,860 minutes, and 6,612 parts.
The method is: take 354 days and 940 minutes, then 348; multiply by thirteen degrees, nineteen minutes, and seven parts after the moon to get the numerator. Also multiply the denominator of degrees by the denominator of days to get the denominator. The product divided by the denominator gives 4,737 degrees, 7,860 minutes, and 6,612 parts. Divide this by 365 degrees, 7,860 minutes, and 4,465 parts of the full circle. The remainder, 354 degrees, 7,860 minutes, and 6,612 parts, is discarded as the month is insufficient; all other values are retained.
For a major year, since the month is insufficient, discard 18 degrees, 7,860 minutes, and 11,628 parts.
The method is: take 383 days and 940 minutes, then 847; multiply by thirteen degrees, nineteen minutes, and seven parts after the moon to get the numerator. Multiply the denominator of degrees by the denominator of days to get the denominator. The product divided by the denominator gives 5,132 degrees, 7,860 minutes, and 2,698 parts. Divide this by the full circle; the remainder is discarded as the month is insufficient.
For a common year, since the month is insufficient, discard 134 degrees, 7,860 minutes, and 115 parts.
The method is: take 365 days and 940 minutes, then 235; multiply by thirteen degrees, nineteen minutes, and seven parts after the moon to get the numerator. Multiply the denominator of degrees by the denominator of days to get the denominator. The product divided by the denominator gives 4,882 degrees, 7,860 minutes, and 14,570 parts. Divide this by the full circle; the remainder is discarded.

The number of degrees by which this month falls short of the standard value is as follows:
For a small month, it is 22 degrees, 7,755 out of 17,860 degrees.
The method is: set 29 days for the small month; multiply by 13 degrees and 19 minutes, divided by 7, to get the numerator. Multiply the degree denominator by the day denominator to get the denominator. The product divided by the full circle gives 387 degrees, 12,220 out of 17,860 degrees. Divide this by the full circle; the remainder is the number of degrees by which the month falls short.
For a large month, it is 35 degrees, 14,335 out of 17,860 degrees.
The method is: set 30 days for the large month; multiply by 13 degrees and 19 minutes, divided by 7, to get the numerator. Multiply the degree denominator by the day denominator to get the denominator. The product divided by the full circle gives 401 degrees, 940 out of 17,860 degrees. Divide this by the full circle; the remainder is the number of degrees by which the month falls short.
For an intercalary month, it is 29 degrees, 9,481 out of 17,860 degrees.
The method is: set 29 days and 490 days for the intercalary month; multiply by 13 degrees and 19 minutes, divided by 7, to get the numerator. Multiply the degree denominator by the day denominator to get the denominator. The product divided by the full circle gives 394 degrees, 13,946 out of 17,860 degrees. Divide this by the full circle; the remainder is the number of degrees by which the month falls short.
Divide by 6,520,3365 to get one full circle; the remainder is 527,421 degrees, which is the amount by which it falls short. Divide this by 17,860 to get 29 degrees for the intercalary month; the remainder is 9,481 degrees, which is then handled using the specific formula.

Volume 3 of Zhou Bi Suan Jing
On the Winter Solstice, the days are extremely short. The sun rises at Chen and sets at Shen. The sun’s rays cover three parts but not nine. East and west are equal; it is due south.
On the Summer Solstice, the days are extremely long. The sun rises at Yin and sets at Xu. The sun’s rays cover nine parts but not three. East and west are equal; it is due north.
The sun rises on the left and sets on the right, moving from north to south.
Therefore, on the Winter Solstice, yang energy follows Kan and is in Zi; the sun rises at Xun and sets at Kun. With little sunlight, it is called cold.
On the Summer Solstice, yin energy follows Li and is in Wu; the sun rises at Gen and sets at Qian. With much sunlight, it is called hot.
When the sun and moon deviate from their proper paths, cold and heat become extreme.
What has passed is false; what comes is true. Thus, falsehood and truth influence each other.
After the Winter Solstice, the sun moves to the right; after the Summer Solstice, it moves to the left. Leftward movement signifies departure, while rightward movement signifies arrival.
When the moon and sun align, it forms one month.
Days repeated form one day.
Days repeated along with the stars form one year.
The external balance occurs at the Winter Solstice; the internal balance occurs at the Summer Solstice.
The six qi cycles back, all referred to as central qi.
The numbers of yin and yang, along with the laws of the sun and moon—nineteen years make one chapter. Four chapters make one zou, totaling seventy-six years. Twenty zous make one suì, totaling 1,520 years. Three suì make one shǒu, totaling 4,560 years. Seven shǒu make one jí, totaling 31,920 years. All cycles come to an end, and all things begin anew.
Heaven uses cyclical changes to mark time.
It is known that heaven has 365 degrees and one-fourth of a degree, and the sun moves one degree per day. The moon lags behind by thirteen degrees and nineteen-sevenths of a degree. 29 days equal 499 out of 940 days, forming one month. Twelve months divided by nineteen equal one year.

Divide by the circumference of the sky.
For those that cannot be divided, such as during a new moon, in ancient times, Shennong created calendars. At the beginning of each era, when the three lights were observed, they did not match the standard.
The sun and moon, along with the stars, had no defined degrees.
The sun represented day, the moon represented night; day and night together made one day. Both the sun and moon rose at the same time relative to the fixed stars.
The moon moved quickly, while the sun moved slowly.
The sun and moon chased each other over a period of twenty-nine to thirty days, during which the sun traveled more than twenty-nine degrees across the sky.
There was no fixed division yet.
Thus, over 365 days, the shadow at the South Pole became longer; the next day it became shorter again. By the end of the year, the shadow became longer once more. Hence, it is known that among 365 days, there are three such periods, and one such period among 366 days.
Therefore, one year consists of 365 days and one-fourth of a day. At the end of the year, the moon has accumulated thirteen cycles beyond the natural cycle, plus an additional amount of more than 134 degrees.
There is also an uncertainty of thirteen degrees and nineteen minutes and seven parts.
Thus, the sun travels 76 cycles across the sky in one year, while the moon travels 1,116 cycles. When they align with the fixed stars, using the number of months’ movement beyond the natural cycle divided by the number of days’ movement beyond the natural cycle gives thirteen degrees, nineteen minutes, and seven parts—this is the degree the moon travels in one day.
Again, using the total number of months over 76 years, dividing by 76 gives twelve months and nineteen minutes and seven parts—this is the number of months in one year.
Using the total degrees of the circumference of the sky, dividing by twelve months and nineteen minutes and seven parts gives 29 days and 940/499 days—this is the number of days in one month.

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