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The Angle Systems in Kao Gong Ji

Guan Zengjian (Zhengzhou University)

Since there was no 360deg central angle division system in ancient China, how did the ancient people express the concept of angles when coming to them? Through detailed studies we find some stipulations of the angle systems used as technical standards in Kao Gong Ji.

There was already some abstract concept of angle in Kao Gong Ji. The ancient people often used the term Jugou (惣�y) to stand for any angle. For example, "Jugou equals one and half a Ju 啿", "Jugou is equal to Qingzhe (鷦菙)", "Jugou is as large as one Ju" and so on. Some people think of Jugou as a specific angle which equals 135deg in the 360deg division system, but the opinion is wrong.

The specific angles frequently used in Kao Gong Ji are Ju and Qingzhe. Ju stands for what is now called the right angle and Qingzhe can be obtained by performing geometrical operations on Ju. The order of the operations described in Kao Gong Ji is as follows: "Half a Ju is called a Xuan (�), one and half a Xuan is the equivalent of one Zhu (耙), one and half a Zhu equals one Ke (灶), and one and half a Ke is one Qingzhe." When explaining those terms, Zheng Xuan of the East Han Dynasty and Kong Yingda of the Tang Dynasty both thought them as referring to length, but they were wrong. In fact those terms form an angle system in which Ju and Qingzhe are most important. Xuan, Zhu and Ke, through which Qingzhe is obtained, play a role of transition. The system is constructible and any specific angle in it can be obtained by performing geometric operations on Ju.

In addition to Ju, Kao Gong Ji also takes Gui (条) as a basis in carrying out geometric operations to obtain some other specific angles which can be used as technical standards for some special purposes.

The features of the angle systems in Kao Gong Ji can be summarized as follows: There exists in the book some abstract concept of angle. There are also some specific angles used as technical standards. These angles are constructed by performing geometric operations on Gui or Ju. That is to say, the angles in Kao Gong Ji are constructible.

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