China's agriculture and handicrafts developed quite early and matured. Advanced agriculture and handicrafts have brought advanced technology, many of which contain graphic knowledge. Including the manufacture and use of surveying and mapping tools, the expression of graphic concepts, the calculation of the equal plane area of land and the three-dimensional volume of granary, etc.
Geometry knowledge in ancient Chinese mathematics has an inherent logic, which is based on practical materials to organize the knowledge system and the calculation of graphics as the central content of knowledge.
the invention of early measuring tools, such as rules and moments, has a direct impact on promoting the development of measurement technology in China.
when Dayu was managing water, he traveled by car on land, by boat on water, by sledge on mud, and by wearing spikes on mountain, which was very hard. He held the yardstick in his left hand and the rules in his right. The Yellow River and the Yangtze River ran around and investigated everywhere.
In order to control water, Dayu walked under the treetops, and his hat was blown off by the branches, and he didn't look back. I lost my shoes and didn't go back to pick them up. In fact, it's not that he doesn't know that his shoes are lost. He won't take the time to pick them up.
Just as there is a famous saying that inspires people, Dayu doesn't like a foot-long jade, but cherishes an inch-long time.
the "quasi", "rope", "gauge" and "moment" in Dayu's hand are the drawing tools in ancient China.
the original drawing must be freehand. With the improvement of graphic requirements, especially the requirements of graphic standardization, such as straight lines and round arcs, the creation of drawing tools has become inevitable.
The style of "quasi" is somewhat like the current T-square. From the word meaning analysis, its function is probably to combine with the rope to determine the straightness of a large range of lines.
The functions of "rules" and "moments" are drawing and setting right angles respectively. These two words have appeared in Oracle Bone Inscriptions. The gauge is taken from the way of holding the gauge by hand, and the moment is taken from its actual shape. The shape of the moment later changed, from containing two right angles to containing only one right angle.
the invention of rules, moments, standards and ropes has a process of gradual formation and perfection in practice. The emergence of these drawing tools has strongly promoted the development of related production, and also greatly enriched and developed people's graphic concepts and geometric knowledge.
during the warring States period, a good technical plan has appeared. The drawings of ships, weapons and buildings drawn on some lacquerware conform to the principle of orthographic projection. There is an architectural plan on a piece of copper in the ancient tomb of Sun Yat-sen during the Warring States Period unearthed in Hebei Province, which shows high drawing skills and geometric level.
the invention of early measuring tools, such as rules and moments, has a direct impact on promoting the development of measurement technology in China.
during the Qin and Han dynasties, measuring tools became more specialized and refined. In order to measure the length, a pole and a measuring rope were invented. The former was used to measure the short distance, while the latter was used to measure the long distance. There is also a soft ruler made of bamboo sticks, which is similar in length to a tape measure. Moment has also developed from scaleless to graduated square.
In addition, a level, a level ruler and a compass were invented. The measurement method is naturally more sophisticated, which can not only measure the reachable targets, but also measure the unreachable targets.
after the Qin and Han dynasties, the brilliancy of measurement methods brought the superb calculation after measurement, thus enriching the content of mathematics in China.
According to Zhou Pian Shu Jing, which was written in the 1st century BC, during the founding of the Western Zhou Dynasty, Duke Zhou and Shang Gao discussed the method of measuring moments, among which the method of using moments mentioned by Shang Gao included rich mathematical contents.
Shang Gao said: "The flat moment is the right rope, the flat moment is the height, the complex moment is the depth, and the lying moment is the distance ..." Shang Gao said that the height, depth and breadth of the target can be measured by placing the square in different positions.
what Shang Gao said about using moments is actually the so-called pythagorean measurement. Pythagorean measurement involves Pythagorean theorem, therefore, the Zhouyi Shujing gives examples of Gousan, Gusi and Xianwu in particular.
after the Qin and Han dynasties, some people wrote books and discussed in detail the method of measuring by using the similarity principle of right triangle. These famous works include Zhou Piai Suan Jing, Nine Chapters Arithmetic, Island Arithmetic Classic, Notes on Arithmetic, Nine Chapters Arithmetic, Siyuan Yujian, etc., which constitute the unique measurement theory of ancient Chinese mathematics.
the concept of graphics is formed in people's practice of contacting and transforming nature. In the early days, human beings obtained graphic knowledge by directly observing nature and imitating nature.
The so-called nature here is not the nature that is generally explained, but the nature that focuses on food according to the most urgent needs of human beings. People get knowledge about animal habits and plant properties from this aspect, and turn from prayer to worship.
Almost all worship methods show the characteristics of primitive art, such as animal dances and murals. It can be believed that we really rely on biological factors in primitive life to use some graphic techniques. This is not only the source of visual art, but also the source of graphic symbols, mathematics and book contracts. With the deepening of life and production practice, the concept of graphics has been strengthened and developed for two main reasons.
First, there are full-time people who use graphics to express people's thoughts and feelings. Judging from the evidence of funerals and murals at the end of the Paleolithic period, it seems that at that time, illusion was already emphasized, and graphics were used as part of the expression of illusion.
Illusion requires full-time staff to perform it. They not only preside over major ceremonies, but also act as painters. In this way, through the work of painters, the style of graphics gradually changes from direct portrait to simplified idols and symbols, which has abstract significance.
the second is the decisive influence of production practice. One of the practical foundations of graphic geometry is weaving. According to textual research, the method of basket weaving was indeed mastered in the Paleolithic Age, and the coarse weaving method appeared in its application.
Weaving is both technology and art. Therefore, besides the general technical laws, there is also artistic beauty to be explored, and both of them must be realized through practice and then thinking. This laid the foundation for geometry and arithmetic.
Because the various forms of woven patterns and the number of longitude and latitude lines they contain are all mathematical in nature, it has aroused a deeper understanding of some relations between shape and number.
Of course, the reason for geometric figures is not only weaving, but also the use of wheels, the construction of brick houses and the measurement of land, which have directly deepened and expanded the understanding of geometric figures and become the basic subject to arouse the ancients to establish geometric concepts.
If we say that the above production practices have made people produce and deepen the concept of graphics, then the drawing of pottery patterns is the occasion for people to show this concept. In the drawing of various patterns, especially geometric patterns, people have once again developed the spatial relationship, which is the mutual position relationship and size relationship between graphics.
Archaeologists' archaeological discoveries have confirmed that as early as the Neolithic Age, ancient Chinese people had obvious concepts of geometric figures. Geometric figures such as diagonal lines, circles, squares, triangles and bisected squares have appeared in the configuration of Banpo site in Xi 'an and the unearthed pottery.
In the triangle drawn, there are different shapes such as right angle, isosceles and equilateral.
The pottery in the later period shows a developed graphic concept. For example, various symmetrical graphics have appeared on the pottery pot unearthed in Pi County, Jiangsu Province. The pattern along the mouth of the pottery basin unearthed from the Xiapanwang site in Cixian County shows the flower teeth with equal circumference. There are almost no regular geometric shapes in nature, but people have created more or less regular objects through practical activities such as weaving and pottery making. These products, which are constantly appearing and handed down from generation to generation, provide an opportunity to compare them with each other, so that people can finally find out the similarities between them and form geometric figures in the abstract sense. The concepts of various geometric figures we have today are first determined by seeing objects with these shapes made by people, and we know how to make them ourselves. In fact, this is also an example of practical knowledge. In ancient China, we also had a certain understanding of diagonal and could apply it. According to Kao Gong Ji, a book written in the Warring States Period, people had already understood and applied the concept of diagonal in the work of manufacturing farm tools, vehicles, weapons and musical instruments. "Zhou Li? According to Kao Gong Ji, craftsmen at that time made farm tools, vehicles, etc., "half a moment means declaration, half a declaration means it, half a ke means it, and half a ke means it." Where "moment" refers to a right angle, that is, 9 degrees. From this calculation, the "first announcement" is 45 degrees, the first one is 67.5 degrees, the first one is 11 degrees and 15 minutes, and the first one is 151 degrees and 52.5 minutes. But this is not very exact. Because in the same book, the size of the "parallel fold" is also said to be "half a moment", so it should be 135 degrees.
The appearance of special names of various corners not only shows the understanding and application of diagonal in handicraft technology, but also reflects the importance of the mathematical significance of diagonal in ancient China. It makes Chinese ancient mathematics solve the problems in practice in another way.
As for the knowledge of area and volume calculation, it is directly related to the establishment of ancient tax system and the perfection of measurement system.
The Spring and Autumn Annals, an important book in the pre-Qin period, recorded Lu Xuangong's "initial tax mu" and began to collect taxes by mu, with "one out of ten". "Guanzi" also recorded Qi Huangong's "case field and tax". To implement these tax systems, we must first make clear the land area and measure the land clearly, and then tax according to the proportion of mu.
This shows that during the Spring and Autumn Period and the Warring States Period, there were methods for measuring land and calculating area and volume in China.
The calculation method of area and volume in the pre-Qin period was later concentrated in the book Nine Chapters Arithmetic in the Western Han Dynasty, which became one of the important contents of mathematical knowledge.
In addition, it can also be proved by archaeologists in Juyan Han bamboo slips. These achievements have been gradually formed in the early accumulation of mathematical knowledge and become the basis of later area and volume theory.