Introduction

Achieving metrological unity is considered as a rather modern feat, but it had already started to preoccupy the minds of early Chinese societies since first-century [1], and the evolution of measurement systems is an important expression of cultural and technological development [2, 3]. In ancient China, the ruler used for construction activities, such as carpentry, masonry, land measurement, construction is called yingzao chi [4]. Yingzao chi is very subtle, and it has three meanings. Firstly, yingzao chi is a graduated physical ruler, used to show how long one chi (尺) is. Secondly, it is the length of the official chi (guanchi 官尺) of the Board of Works of a Certain Dynasty. Generally, it is called official chi that is promulgated by the official and implemented as the measure standard [5], changing with dynasty regime [6]. In general, the numerical value of official chi shows a trend of continuous increase in history (Table 1). Thirdly, yingzao chi is the chi length unit for a certain construction work with regional characteristics, which stems from but is distinguished from the official chi. Due to the cumulative errors in long-term use, unbalanced regional development, the apprentice secretly copies the master's yingzao chi and other factors [7], yingzao chi is different in different regions so that have possibilities to reflect the regional characteristics [8, 9]. If the yingzao chi of multiple ancient buildings in an area is integrated together, the range of vernacular chi can be obtained (Fig. 1). Li reports that the yingzao chi in the south of China has obvious regional characteristics, thus forming a certain " vernacular chi system", such as the "Zhe chi (浙尺) system" in Zhejiang, the "Min chi (闽尺) system" in Fujian, and the "Yue chi (粤尺) system" in Guangdong, etc. [10], based on literature on Chinese architecture, the oral accounts of craftsmen from all over China, physical yingzao chi unearthed in multiple places and the research results from the team he led at Tongji University. And if the yingzao chi value of several buildings in the same area fluctuates is within 3 mm, the probability that these yingzao chi are homologous is greater than the probability that they are non-homologous [11]. Therefore, as a unit of length with regional characteristics, yingzao chi has the potential to tell us about how an ancient people counted and quantified their world [12, 13]. This study focuses on the yingzao chi under the third meaning.

Table 1 One chi-measure unit to the official chi in some Chinese dynasties (unit: mm)
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Fig. 1
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The relationship between yingzao chi, official chi and vernacular chi

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The length of one chi is different in each place, which directly causes people in different regions to have different perceptions of scale. So, if buildings are built with the same design values, their actual scales are not the same in different regions. Such subtle construction process is fully reflected in Late Imperial Chinese architecture. The palace buildings of Ming and Qing Dynasties is considered as one of the important cultural heritages [14, 15], which can be divided into two types: with dougong and without dougong. Their structural traditions are the basis for the evolution of ancient architectural technology [16,17,18]. Although dougong is able to show the unique exquisite craftsmanship used in high-grade buildings, in fact, the presence of dougong free in palace buildings is not an expression of architectural construction skills deterioration. Conversely, the palace building sans dougong is the evidence of an optimized beam-column frame construction. As a component between beams and columns to reduce the vitality of the building [19, 20], dougong has made a further contribution to the upturned eaves. However, dougong is not obligatory for the design and construction of ancient Chinese buildings. From Tang Dynasty to Qing Dynasty, the structural role of dougong in buildings gradually weakened [21], replaced by the role of decoration. The reason for this phenomenon may be changes in building materials. There was a lack of wood aniseed in Ming and Qing Dynasties, compared to before [22]. Another reason may be that during Ming and Qing Dynasties, rainfall decreased [23], and the materials more waterproof than wood [24], such as brick and stone, were more widely used in buildings [25], so buildings no longer needed such far-reaching eaves. In addition, the construction technology of Chinese palace buildings has been improving. From Tang Dynasty to Qing Dynasty, ancient Chinese people have been exploring more precise beam-column frame construction in buildings, which simplified the joints between beams and columns (such as dougong) (Fig. 2), and made it possible to design and build more space-using buildings with limited materials. Liang Sicheng shows the evolution of dougong in A Pictorial History of Chinese Architecture. Based on the field research, historical materials, and published books [25, 26], the comparison of three palace building sections with double-eaved gable-and-hip roof but different construction techniques in Chongqing is shown in Table 2.

Fig. 2
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Dougong evolution diagram

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Table 2 Comparison of palace building sections with different construction techniques in Chongqing
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As a special symbol of the Chinese construction system and the inheritance of craftsman school,Footnote 1yingzao chi provides a perspective for us to study the ancient architecture system with local characteristics, clarify different craftsman school systems, and achieve the ultimate goal of genealogy construction and reginal division. However, the loss of yingzao chi creates substantial challenge for this great goal. In the absence of physical objects of yingzao chi, the formula from the craftsman and the literature description, the value of yingzao chi cannot be directly obtained, and can only be restored according to the Chinese architectural scale design theory, which is very difficult. Past research has found that the basic dimensions of a building mainly depend on the scale of the human body and the scale of the building components [27, 28]. In this regard, ancient Chinese people from Tang Dynasty to Qing Dynasty have two kinds of architectural design concept. The first one is called yingzao chi theory, which aims to design the absolute dimensions of building widths, depths and column heights by directly assigning absolute values in the unit of yingzao chi [28, 29]. This theory focuses more on the design of architectural space. The other one is called modulus theory, which aims to use objects such as dougong, material (cai 材) or other building components as reference objects to design the dimensions of the building cross-sections, post network dimensions and heights [30]. This theory focuses more on the design of timber frames. For Tang and Song dynasty buildings, as well as Ming and Qing Dynasty buildings with dougong, these two kinds of concept can complement each other and should be combined in the design of ancient buildings. And based on a large number of examples, the following viewpoints are analyzed and summarized (Fig. 3): if the dimensions of building widths and depths are measured in the unit of yingzao chi, the design values of these dimensions will be integers or half integers, and there is also a possibility of quarter integers. This rule is called "integer system" (整尺制) and " bisection system" (对分制) [11]. Especially for palace buildings sans dougong in late-Imperial period, of which the scale design is directly based on the yingzao chi [31], these two systems are very obvious. Such approach allows for more flexibility in the design values of the building scale, which also leads to more sensitivity to “how long one chi is”. While if a reference object, the size of which is related to the value of the yingzao chi [11, 32] is used as a module to measure the cross-section, height, and other scales of the components in the building timber frame, the design values will be integers. And this rule is called "full-doubling system" (整倍制) [11], which is shown evidently in palace buildings with dougong. This is a method that relies on relative proportions, choosing a more appropriate reference than "chi" to measure the scale of a building. Xiao is devoted to the study of “basic module theory" in the scale design of Chinese ancient architecture [4], supporting the concept of "full-doubling system" greatly. He defines the maximum common divisor of the post network dimensions as the basic module of the building, successfully applies the "basic module theory" to the scale analysis of ancient buildings in Tang and Song Dynasties and concludes that the numerical value of the basic module of such buildings is between 0.5 and 1 times the numerical value of the yingzao chi based on a large number of cases [4].

Fig. 3
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Architectural scale design theory

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According to these theories, there are two main methods to restore yingzao chi: (1) First, the greatest common divisor of post network dimensions of the building is derived, which is determined to be the basic module M of the building. And M can locate the column network position in integer multiples. Then according to the "M values between 5–10 inches" theory, M is divided by the integer and half-integer between 5–10 respectively, to get the potential yingzao chi value (Y1, Y2, Y3…). Finally, these results are compared with the known vernacular chi value, and the nearest value is taken as the restored value of the yingzao chi [33]. This method is applicable to the ancient buildings in Tang and Song Dynasties. (2) First, the post network dimensions are divided by the official chi or vernacular chi (Y1, Y2, Y3…) to obtain the potential restored design value in the unit of yingzao chi. Finally, the yingzao chi that can restore the most integer-design values is regarded as the restored yingzao chi [34]. This method is applicable to the palace buidings with dougong in Ming and Qing Dynasties. There is no doubt that the previous methods rely to a large extent on the official chi or the vernacular chi with a small range of values as an effective reference when restoring the yingzao chi.

However, not every vernacular chi has been well studied. For example, there is little research on vernacular chi in Chongqing, with no specific conclusion on the value of vernacular chi at present. Only two cases have been published: the first one is Longxing Ancient Town, of which the yingzao chi is 372 mm [10], and the second one is Dacheng Hall in Chongqing Literature Temple, of which the yingzao chi is 326.7 mm [26]. Even though the reference range is expanded to the Sichuan-Chongqing area, the current known yingzao chi value range is 318–400 mm [10], which is not effective enough. Therefore, where there are no thorough research results of vernacular chi, the previous yingzao chi restoration methods cannot be used. Secondly, the previous methods only consider the conditions of integer and half-integer building design values in the unit of yingzao chi, but not consider the conditions of quarters, directly ignoring the error in the process of deriving common diversions from building surveying data. In addition, the theory about the basic modulus M is a result summarized based on the building scale data of Yuan Dynasty or earlier [4], which may not be applicable to the buildings in Ming and Qing Dynasties.

Based on the above phenomena, this paper taking Chongqing as a typical example where there is a lack of sound research on vernacular chi, studies and improves a theoretical method to help restore the yingzao chi of palace buildings without dougong in the Ming and Qing Dynasties (Table 3). This method can restore the yingzao chi of a single building through surveying data, and is not limited by the lack of reference of vernacular chi. Moreover, the high similarity of the restored yingzao chi of several buildings of the same period and type in the same area verifies the reliability of the method and results.

Table 3 Comparison of the yingzao chi restoration methods in this study and in the previous
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Research methodology

This study concludes that the yingzao chi theory and modulus theory can exist in the scale analysis of ancient architecture at the same time, and there is a multiple relation between them. The yingzao chi theory states that yingzao chi can control the spatial scale design values (such as widths and depths) of the building in integer, half, or quarter times. While modulus theory explains that the modulus of the building can be used to control the structure design values of the building (such as post network dimensions, the height of the beam from the ground, etc.) and the design value of the section size of components (such as the column diameter and the thickness of the beam, etc.) by integer multiple. In fact, the values of widths and depths of the building are equal to the values of post network dimensions. And the modulus theory is evident in palace buildings with dougong, but the yingzao chi restoration of palace buildings without dougong does not need to rely on modulus. In this study, three typical palace buildings without dougong in Chongqing during the Ming and Qing Dynasties are selected as examples for analysis, and the yingzao chi restoration method is used respectively, as shown in Fig. 4.

Fig. 4
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The steps of restoring the yingzao chi of palace buildings without dougong in Ming and Qing Dynasties

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The first step to restore the yingzao chi is to find the common divisors of the surveying data of post network dimensions, including the dimensions of the widths, depths, and corridors. Based on the "integer system" and " bisection system", the surveying data of post network dimensions of three typical palace buildings without dougong in Chongqing is divided by the integers, half integers and quarter integers between 4 and 19, respectively. After screening, the average of similar results is used as the potential yingzao chi value. The screening indicators are as follows: (1) Only the values in the range of "318 mm–400 mm" are screened. (2) The values used for averaging are as close as possible, with a maximum difference of 10 mm. Under this criterion, if the three buildings have very close potential yingzao chi in value, and the maximum difference between these values is within 3 mm, these yingzao chi are identified as the restored yingzao chi of each of the three buildings. Because under such indicators, these yingzao chi have a high possibility of homology, and their similarity can reflect the vernacular chi in Chongqing. And then the following research objectives will be realized:

(1) Based on the construction and yingzao chi theory of ancient Chinese architecture, a theoretical method is designed to help restore the yingzao chi of palace buildings without dougong in the Ming and Qing Dynasties.

(2) Based on the post network dimensions data of three typical buildings without dougong in Chongqing during the Ming and Qing Dynasties, the research method designed in this paper is used to analyze and speculate the numerical range of vernacular chi in Chongqing.

(3) Combined with the known yingzao chi values of Chongqing ancient buildings, their types and historical origins, the errors between the restored dimensions and the surveying data are calculated respectively, and the most likely yingzao chi values of these three cases are deduced.

Case analysis

The existing palace buildings without dougong in Ming and Qing Dynasties in Chongqing are very scarce. According to records, there are 23 temples or Taoist temples in Chongqing, and only 4 temples have this type of architecture preserved [25]. The three typical palace buildings without dougong in Ming and Qing Dynasties selected in this paper are the Main Hall of Huayan Temple, Guansheng Hall of Shuang-Gui-Tang and the Main Hall of Shengshou Temple (Table 4). Moreover, the types of these three buildings have been identified again through field research, historical materials and published books [35, 36]. Judging from the age of construction and structural form, it is possible that these three buildings were built by craftsmen from the same school. The study of their yingzao chi and the architectural scale rule has high historical, artistic and scientific value, and become an important bridge that maintains the cultural connection between ancient and modern humans [37].

Table 4 Comparison of the cases selected in this paper and the cases that have determined the value of yingzao chi
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Huayan Temple is a typical representative of Chongqing Buddhist buildings, located in Huayan Township, Jiulongpo District, Chongqing. The Main Hall of Huayan Temple was built in the Ming Dynasty and rebuilt in the 31st year of Guangxu of the Qing Dynasty. It is one of the core buildings of the whole temple [25, 38] with mixed beam and bucket beam structure, double-eaved gable-and-hip roof and a spatial pattern of 5 rooms on the wide side and 4 rooms on the deep side. Although the appearance of the Main Hall has changed during the previous restorations, the main body structure is still unchanged. Based on the historical data before the ceiling decoration, on-site hard ruler measurement data, and 3D laser scanner measurement results, the surveying and mapping group organized by the School of Architecture and Urban Planning Chongqing University redraws the current plan (Fig. 5) and section diagram of the Main Hall of Huayan Temple.

Fig. 5
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Plan of the Main Hall of Huayan Temple

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Shuang-Gui-Tang is located in Liangping County, Chongqing. The Guansheng Hall of Shuang-Gui-Tang was built in the tenth year of Qing Shunzhi (1653) and rebuilt in the 44th year of Qing Qianlong (1779). It has three rooms on the wide side, with a single-eaved gable-and-hip roof. Its structure is a combination of stone and wood frame with brick walls [36]. In this paper, the dimensions of widths and depths of the building are obtained from the plan and the marked post network dimensions published in Liangping Shuang-Gui-Tang (Fig. 6).

Fig. 6
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Plan of the Guansheng Hall of Shuang-Gui-Tang

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Shengshou Temple is located in Dazu County, Chongqing. The Main Hall of Shengshou Temple was rebuilt during the Kangxi period of the Qing Dynasty (1662–1722), and repaired in the first year of Tongzhi (1862). It has five rooms on the wide side, and a double-eaved gable-and-hip roof [35]. In this paper, the dimensions of widths and depths of the building are obtained from the plan and the marked post network dimensions published in Dazu stone carvings and ancient buildings (Fig. 7).

Fig. 7
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Plan of the Main Hall of the Main Hall of Shengshou Temple

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Derivation of yingzao chi

The central axis of the column is used as the positioning line of the column network, and the post network dimensions are equal to the widths and depths of the building. First, the post network dimensions data of the three buildings are extracted and divided by the integers, half integers, and quarter integers between 4 and 19, respectively, keeping the results to one decimal place (Tables 5, 6, 7). After sifting, the average of similar results is taken as the common divisors of these dimensions data and is identified as the potential yingzao chi value (Figs. 8, 9, 10).

Table 5 Main scale parameters of the Main Hall of Huayan Temple (unit: mm)
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Table 6 Main scale parameters of the Guansheng Hall of Shuang-Gui-Tang (unit: mm)
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Table 7 Main scale parameters of the Main Hall of Shengshou Temple (unit: mm)
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Fig. 8
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Process of deriving common divisors from post network dimensions data of the Main Hall of Huayan Temple

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Fig. 9
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Process of deriving common divisors from post network dimensions data of the Guansheng Hall of Shuang-Gui-Tang

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Fig. 10
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Process of deriving common divisors from post network dimensions data of the Main Hall of Shengshou Temple

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The results show that the Main Hall of Huayan Temple, the Guansheng Hall of Shuang-Gui-Tang and the Main Hall of Shengshou Temple all have five different values of yingzao chi. Through comparative analysis of these 5 groups of potential yingzao chi, it is found that these three buildings have two groups of potential yingzao chi (Y4 and Y5) that are very close in value, and their fluctuation range is within 3 mm (Table 8). Therefore, it can be basically determined that their yingzao chi is Y4 (334.2–336.3 mm) or Y5 (320.7–323.6 mm).

Table 8 Comparative analysis of potential yingzao chi for these three buildings (unit: mm)
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Discussion

As for the three buildings as case studies in this paper, this paper believes that Y5 is more likely to be the restored yingzao chi. There are two reasons: (1) The yingzao chi of Dacheng Hall in Chongqing Literature Temple is 326.7 mm, which is very close to Y5. In addition, the construction periods of the three cases studied in this paper are similar to that of Dacheng Hall in Chongqing Literature Temple, and these buildings are all palace buildings. (2) If the building restoration size error analysis based on these two groups of potential yingzao chi (Y4 and Y5) is used to select a more ideal yingzao chi value, the result will support Y5 (Tables 9, 10, 11). The specific step is to divide the surveying data by the potential values of yingzao chi (Y4 and Y5), and adjust the dimension values in the unit of yingzao chi according to the "integer system" and " bisection system". The adjusted values are the restored dimensions, and then the error between the restored dimensions and the surveying data is calculated. The results show that first, the error of the dimensions restored by Y5 is less. Second, Y5 can be used to recover more integral or half-integer building dimensions. This phenomenon can be seen in all three buildings, with almost all the error of less than 1%, which is high accuracy for the ancient timber structure that built in the seventeenth or eighteenth century.

Table 9 Error analysis of dimension restoration of the Main Hall of Huayan Temple based on Y4 and Y5
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Table 10 Error analysis of dimension restoration of the Guansheng Hall of Shuang-Gui-Tang based on Y4 and Y5
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Table 11 Error analysis of dimension restoration of the Main Hall of Shengshou Temple based on Y4 and Y5
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It is very important to combine the comparative analysis of buildings of the same period and type in the same area for the yingzao chi restoration, which cannot only be verified by the dimension restoration error. Through the calculation and analysis, all the potential yingzao chi of each building can be used to restore a small error dimension, taking the Main Hall of Huayan Temple as an example below (Table 12). This is because the sample selected in this paper is comprehensive, including all dimensions of width and depth. Moreover, in the process of deriving the common divisors, each dimension data is divided by integers, half integers and quarter integers between 4 and 19 respectively to help reduce the error in the process of deriving the common divisors. So compared with the previous sample survey, or the method only dividing integers and half integers between 4 and 19, the complete process in this study can provide more credible results [39, 40]. In addition, in the screening process of potential yingzao chi based on the post network dimensions, it is found that the smaller dimension data is more critical to determine the approximate range of yingzao chi. Because when each data is divided by integers, half integers, and quarter integers between 4 and 19, the larger dimension data can always yield a lot of similar results.

Table 12 Error analysis of dimension restoration of the Main Hall of Huayan Temple based on Y1, Y2 and Y3
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However, although the sizes of building structural components are also designed in the unit of the yingzao chi, it is invalid to use the section sizes of building structural components to screen the most likely yingzao chi. This is because the section size of building structural components itself is very small, coupled with the long-term deformation of wood [41], measurement errors [42] and other influencing factors. Taking the Main Hall of Huayan Temple as an example below (Table 13), by dividing the sectional dimensions of the structural components by Y4 and Y5 respectively, it is found that these results are close.

Table 13 Sectional dimensions of the structural components of the Main Hall of Huayan Temple
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In addition, in terms of the relationship between plane scale and height, the dimensions in the width and depth of the building can correspond to that in the key nodes in sections and facades (Figs. 11, 12, 13). This shows that although the modulus system is not obvious in the scale design of such buildings, the building height may be designed according to the scale of the building plane. And for these three buildings, the larger the dimension ratio of the outermost width to the sub-outer width is, the larger the dimension ratio of the height to the half through-surface width will be (Table 14). The characteristic forms of such a height-width ratio may not only have been produced with an aesthetic purpose but were also intended to serve a utilitarian function, especially for structural stability [43]. The larger the spatial scale of the building is, the more numerous and thicker the beams will be, which are the horizontal structure upheld by the columns, and also called ling (檩) or fang (枋) in late Imperial Chinese architecture. The cause for such phenomenon can explain that the aesthetic value of the edifices conforms to the rules of historical architectural theory and practice [44,45,46]. From the appearance of the building, the Main Hall of Huayan Temple is very similar to the Main Hall of Shengshou Temple, while the architectural style of the Guansheng Hall of Shuang-Gui-Tang is obviously different from that of the others. If the yingzao chi of these three buildings is homologous, it is quite possible that they were built by craftsmen from the same school, and the reasons for their different architectural styles are worth exploring. For example, the Guansheng Hall is of a lower building grade than the Main Hall. However, if these three buildings were not built by craftsmen from the same school, then their yingzao chi may not be of the same origin. In this case, to determine how long one chi-measure unit to the yingzao chi of these three buildings is, it is necessary to combine the yingzao chi of other buildings of the same origin with them. After all, yingzao chi is the necessary tool for every craftsman to build buildings.

Fig. 11
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Section view of the Main Hall of Huayan Temple

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Fig. 12
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Section view of the Guansheng Hall of Shuang-Gui-Tang

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Fig. 13
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Section view of the Main Hall of Shengshou Temple

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Table 14 The proportional relationship between the width and height of the building
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The general rules for the construction of palace buildings in the Tang, Song, Ming and Qing dynasties are documented in the imperially commissioned Northern Song building manual Yingzao fashi 營造法式 (Building standards; published in 1103) and the monograph on Qing dynasty architecture by Liang Sicheng Qing Structural Regulations 清代营造则例 (Based his research on 1734 Qing dynasty Architecture Method清工程做法则例; first published in 1934). This is why the appearance of ancient Chinese buildings shows a uniform similarity. However, the regulations on the construction of palace buildings in books are theoretical, including the relationship between the sizes of structural components such as beams and columns and the dimensions of buildings widths and depths, which is likely to be based on generally empirical values. Beyond that, underlying the comprehensive building methods, elaborate structural or decorative details and strict regulations on building materials and manpower is a rich store of distinctive cultural connotations [47]. This is why after designing the building scale based on construction specifications, the construction details on the site are characteristic in the different regions. In fact, building materials, construction levels, and construction scales are all different in each region, which results in the characteristic construction details of local buildings, including the shape and structure of the building. So, in the domain of building knowledge, scholars and craftsmen learn from each other and promote each other, which uniquely yet significantly contributed to practical and theoretical architectural knowledge, an essential aspect of Chinese civilization [48]. As for Chongqing, the warping of gable-and-hip roof is not as gentle as that of northern buildings, but has the characteristics of wing angle (翼角) shape of the buildings in regions south of the Yangtze River, with high warping and beautiful curves. In order to create such a shape, in the wing angle structure, small corner beam (仔角梁) is supported by big corner beam (老角梁), and both sides of the corner beam are supported by "shrimp-antennas wood (虾须木)" to maintain the stability between the components. The shrimp-antennas wood is a unique structural member in Chongqing area. It is a curved round wood with one end connected to the small corner beam and the other end nailed to the skin of the eave purlin (Fig. 14). Such unique wing angle construction method developed locally in Chongqing has been applied to buildings in most areas of Chongqing, regardless of their scale and the age of construction [25].

Fig. 14
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Wing angle upward view of the Main Hall of Huayan Temple

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The practice of ancient architecture with local characteristics is a process passed down from generation to generation. When performing construction activities according to the official architectural design manual, the craftsmen in different regions are bound to face different difficulties due to the local materials, climate or terrain. After many attempts, the craftsman will get some construction experience and tell these experiences to their apprentices. So, such subtle architecture with the craftsman school lineage may reflect the wisdom of local craftsmen to adapt to local conditions based on official standards in terms of form, aesthetics, and technology. However, under the background of the strong centralization of ancient Chinese feudal dynasties, it is very unlikely that local craftsmen deliberately adjusted the length of the yingzao chi for the purpose of "novelty". Instead, the reason for the difference between vernacular chi and official chi is more likely to be a kind of political power product, or a deviation or misunderstanding during the propagation of yingzao chi, such as the wear or deformation of the physical yingzao chi, the error when describing length, and the yingzao chi used in the remote area could not timely be replaced by the official chi of the new dynasty. After all, each craftsman school may be convinced that it has the most orthodox technique. On the whole, the restoration of the yingzao chi provides a potential opportunity to explore the propagation path of aesthetics, technology and culture in ancient China.

Conclusions

This study elucidates the potential numerical values of the measure standards used by the craftsmen of specific buildings in a complex system of entanglement between official and vernacular chi-system. Based on the theories about the scale rule and yingzao chi of Chinese ancient architecture, a theoretical method to help restore the yingzao chi of palace buildings sans dougong in Ming and Qing Dynasties is improved. This method can be effectively applied to the yingzao chi restoration of the palace buildings without dougong in Ming and Qing Dynasties and is not limited by the lack of reference of vernacular chi. After applying this method to the yingzao chi restoration process of three typical cases by their surveying data, it is found that these results have a high degree of homology, which can help to prove the reliability of the method and results in this paper. At present, it can be basically determined that the restored yingzao chi values of the three buildings in this paper are 334.2–336.3 mm or 320.7–323.6 mm, but the latter is more likely. These results can help to improve the study of vernacular chi, understand the scale rule of this kind of architecture, and provide theoretical support for the protection and restoration of such architectural heritage.

This study also has some limitations. The method studied and optimized in this paper to help restore the yingzao chi is not suitable for the case of "Yabai (压白) system" [11]. "Yabai" means that in the scale design of traditional wooden buildings, in order to seek good luck and avoid harm, the dimension value is deliberately made with a specific requirement number, which is common in regions south of the Yangtze River and southeast coastal areas. In addition, the archaeological information about the yingzao chi in Chongqing is limited. The reliability of the results in this paper can be further verified if more ancient books or physical yingzao chi of Ming and Qing Dynasties in Chongqing are unearthed in the future.