There is an old saying that "if you don't follow the rules, you can't make a square circle." Modern people are accustomed to putting "rules" together as a word, but in ancient times, rules are rules, and moments are moments, which are two tools used by the ancients to measure and draw: "moments" are a curved ruler marked with scales folded at right angles, and "rules" are compasses specially used to draw circles.
The invention and application of rules and moments on the mainland began very early, and the words "rules" and "moments" were used in the oracle bones of the 15th century BC. In the records of the "Records of History" on the treatment of water by Dayu, it is also mentioned that Xia Yu used rules and manners to measure and plan the content of the water control plan.
The Zhou Dynasty mathematician Shang Gao once summed up a variety of ways to use the moment: "The flat moment is based on the positive rope, the moment is based on the height, the covering moment is measured by the depth, and the lying moment is known to know the distance." "Flat moment to positive rope" means to place one side of the moment horizontally, the other side is leaning on a plumb line, you can determine whether the rope is vertical; "moment to look high" means to put one side of the moment upside down, you can measure the height; "covering the moment to measure the depth" means that the above-mentioned height measurement moment can be reversed, you can measure the depth; "lying moment to know the distance" means that the above high measurement moment lying flat on the ground, you can measure the distance between the two places. In simple terms, it is to use the different pendulum methods of the moment to determine the horizontal and vertical directions according to the proportional relationship of the corresponding edges of the Pythagorean shape to measure the height, depth and distance of distant objects.
Before the Han Dynasty, the two arms of the curved ruler were equal length and had no scale; the moments of the Warring States period were specially used to draw right angles; after the Han Dynasty, the two arms of the curved ruler were long and short, and with a scale, the moment at this time evolved into a geometric calculation tool. The length of the two arms of the ruler is different, making it easier to hold the operation and read the data.
Among the many mythological figures carved on the bricks and stones of the Han Dynasty, you will see Fuxi holding a moment and a nuwa holding a rule. From the ancient imagery, the structure of the gauge is roughly with two parallel feet, one of which is used to fix the center of the circle and the other foot is used to draw the circle. This kind of compass is very similar to the modern wooden beam compass, and is generally used to draw circles with larger radii.
Ancient China attached great importance to the study of circles and squares, but due to the limitations of the scientific level at that time, it was not a simple task to make accurate measurements of circles in practical applications. So the ancients instead thought of a way to represent and measure circles by connecting squares to circles.
The use of rules and manners greatly promoted the development of ancient geometry on the mainland. In the accounts related to "moments", the most important proposition is the Pythagorean theorem. The Pythagorean theorem is one of the most significant discoveries in the early history of mathematics in the mainland, and there is a record of "therefore folding moments, thinking that the hook is three, the stock repair is four, and the diameter is five", which means that the ratio of the three sides of the Pythagorean shape is 3:4:5, which is a relatively special form of the Pythagorean theorem. In addition, the book also mentions the nature of "ring moments think of circles".
The application of rules and manners enabled the ancients to accurately draw geometric patterns with uniform shapes. Just like the cultural relics we see today, the various shapes of the utensils are because the craftsmen skillfully used the rules and manners, and at the same time used them to draw various geometric patterns such as points, lines, circles, arcs, squares, triangles and so on on.