Abstract
JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 2019, VOL. 18, NO. 5, 457–471 https://doi.org/10.1080/13467581.2019.1680376 ARCHITECTURAL HISTORY AND THEORY Reconsidering a proportional system of timber-frame structures through ancient mathematics books: a case study on the Muryangsujŏn Hall at Pusŏksa Buddhist Monastery Juhwan Cha and Young Jae Kim Department of Heritage Conservation and Restoration, Korea National University of Cultural Heritage, Buyeo, Korea ABSTRACT ARTICLE HISTORY Received 26 December 2018 The mathematics references in ancient China, the Zhoubi Suanjing and the Jiuzhang Suanshu, Accepted 30 September 2019 present information on formative ideas of ancient people and their perception of objects. The introduction to the Yingzao Fashi mentions mathematical sources, including the Zhoubi Suanjing. KEYWORDS Both of these books focus on the philosophical concept of Tianyuan difang (Heaven is round and Muryangsujŏn Hall at Earth is square), as well as inscribed and circumscribed circles. The square root of 2(√2), which can Pusŏksa Buddhist Monastery; be derived from this part, proves to be an essential criterion for building, seen in Korea, China, and √2; Zhoubi Suanjing; Japan. Using the exemplary Koryŏ building, the MuryangsujŏnHall at Pusŏksa Buddhist Jiuzhang Suanshu; Yingzao Fashi Monastery, this thesis shows that the standard ground plan width of the outermost bay has a √2 ratio to the central bay width. Its cross-section, likewise, proves that √2 times or twice the distance or height (relying on the height of the eave columns) are applied to the distance or height between each column and purlin in the application of arithmetic and geometric concepts. In the future, this work will be a reference for the reconstruction design of ancient buildings prior to the Koryŏ period, analogous to the MuryangsujŏnHall. 1. Introduction significance for structural stability. In addition, the scale is an important factor in deciding the size or location of 1.1. Research background a wooden building. The decision regarding the position Studies of timber-frame systems used in East Asian woo- of purlinsand columns isstemmed from thegroundplan. den buildings have been conducted for many years, Depending on historical periods, there have been diverse occupying a very important position in the architectural positions and sizes of beams and girders, as well as history of Korea, China, and Japan. Chinese architecture various methods of integrating bracket complexes to in the Tang, Liao, Jin, and Yuan Dynasties, in common interior framed structures. It appears that there were with that of the Ming and Qing eras, produced architec- certain rules for construction; however, the earliest tural dignities. In Korea, there are three building styles ancient documents that apply to the design concepts of based on a bracket system (wooden structural elements wooden architecture on the Korean Peninsula are records fitted to the tops of columns or beams, in order to of construction following the mid-Chosŏn period. There support the weight of roof eaves), which were prevalent is little information or remaining pieces of buildings dat- in the Koryŏ period (918–1392): 1) the chusimp’o柱心包 ing from earlier times. This is a difficult aspect of the style, which placed the bracket complex directly on the search for formative ideologies or restoration designs in column head, 2) the tap’o多包 style, which included an the Paekche, Silla and Unified Silla periods during the pre- inter-columnar bracket complex besides those on the Koryŏ era. This study, therefore, attempts to derive these column heads, as well as spaces between the brackets concepts from East Asian mathematics texts, which pro- (the dominant architectural style of the Chosŏnperiod), vide a glimpse of ancient formative thinking, as well as and 3) the ik-kong 翼工 style, a bracket complex that the associated figures, and the proportional system uni- featured simplified, beak-like protrusions on highly ted with these figures. important buildings. Such bracket systems have been developed into various shapes over time. 1.2. Existing research problems The ground plan determines structural forms and techniques used to build an interior timber-frame struc- In order to restore ancient Korean architecture and urban ture, and the uses and positions of the inner elements, landscapes of the Silla, Unified Silla, and Paekche periods, such as the purlins, beams, and columns, are of great reconstruction designs rest on the building methods CONTACT Young Jae Kim kyjandy@nuch.ac.kr Division of Architecture · Landscape-architecture · Urbanism, Department of Heritage Conservation and Restoration, Korea National University of Cultural Heritage, Buyeo 33115, Republic of Korea The McCune-Reischauer system of Romanization generally is used throughout this thesis, with some exceptions, especially for earlier common usages of Korean names, such as Cha, Kim, Han, etc. © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the Architectural Institute of Japan, Architectural Institute of Korea and Architectural Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 458 J. CHA AND Y. J. KIM described in the Yingzao Fashi 營造法式 (Treatise on lacking in the connection between the ground plan and Architectural Methods, 1103 CE) and then compared to scale. Dongdadian (the Great East Hall) at Foguangsi other extant Koryŏ buildings. There are few mathematical Monastery, Wutaishan, Shanxi, is among the oldest build- models of the proportional system associated with woo- ings in China and does not follow the 1: √2 rule dictating den architecture, and much doubt has been cast on the the ratio between the height of the eaves columns and reliability of reconstruction designs. However, some the height of the eaves purlins. Recently, Wang Nan’s Chinese researchers lay the groundwork for reconstruc- studies shed new light on the subject. He sketches out tion and restoration designs in the future. First of all, Chen the proportional relationship between square and circle Mingda, referring to the proportions of √2:1 or 3:2, drawings with the ratio of √2 times in that the Zhoubi obtained the ratio of the horizontal and vertical sections Suanjing and Jiuzhang Suanshu include the contents of of the first, balcony substructure, and second floors while the round-square and square-circle maps, and the princi- analysing the example of Dulesi Monastery’sthe ple of “tianyuan difang.” Wang’sresearchesconfirm that Guanyinge Pavilion (Chen 2007, 16).WangGuixianghas the square-circle diagrams with geometric rapport are proposed a valid argument for a 1: √2ratio of theheight commonly applied in designing ancient capital cities of the eaves columns (exterior columns or perimeter (from the Xia to the Qing Dynasties), the layouts of impor- columns, named yanzhu 檐柱 and weizhu 外柱 in tant landmark building complexes for the central axis of Chinese) and the height of the eaves purlins (Wang Beijing, forty-one Buddhist pagodas, the Great East Hall at 2011a, 2011b). (Figure 1)The √2description, along with Monastery Foguangsi, the Guanyinge Hall and the the formative thinking of ancient people, is not enough; Shanmen Gate at Dulesi Buddhist Temple in Liao archi- an explanation of the proportional system of an interior tecture, and palace buildings (the Forbidden City during timber-frame structure remains very vague. Details are Ming-Qing Dynasties). They all embrace the ratio of the Figure 1. A line drawing of proportional system from a Wang Guixiang book cover. Although we do not sure how the Yingzao Fashi has influenced Korean architecture, a few researchers attempt the reconstruction plan assuming the impact on the Yingzao Fashi in Korea. Among them, Kang and Yun proposes a reconstruction plan about the Monastery Pulguksa’sKŭngnakchŏn Hall, drawing on document data from the Yingzao Fashi and archaeological remains about wooden structures left in Korea, China and Japan (Kang and Yun 2006, 217–218). Another application of the Yingzao Fashi has come to focus on the restoration plan of the corridor territories at the Mirŭksa Buddhist Temple, although the monastery’s foundation does not accord with the publication of the Yingzao Fashi. The research accompanied by fellow researchers at the National Research Institute of Cultural Heritage in Korea is animated by two appropriate assumptions. They argue that the restoration plan can produce a compromise through in-depth contemplation to the langwu 廊屋 (roofed corridors linking main buildings) in the Yingzao Fashi’s “Damuzuo zhidu” (the major carpentry system) which contains the proportional system coupled with building grades, and, admittedly, they opine that it can come up with a settlement through the utilization of proportional systems employed for the restoration plans of roofed corridors at other extant Buddhist temples such as Horyuji 法隆寺, Yamadadera 山田寺, and Mikawa Kokubunji 三河國分尼寺 (Kungnip munhwajae yŏn’guso 2018,80–88). Wang’s two articles give a good account for inscribed and circumscribed circles as examples of the Tianyuan difang in ancient literature, as well as the conjunction between 1 and √2 ratio, which are found in the Kanxiang, the beginning section of the Yingzao Fashi. For reference, Fu Xinian, a Chinese scholar, underlines that the concept of geometric centrality has a long history in China through interesting analysis of the modularity and systematic planning of palace and state temple architecture, site layouts. In the choice of the square grid system for a long history which depends on the importance and size of the architectural cluster, Fu argues that Daming Palace and Luoyang Palace in the Tang capitals Chang’an and Luoyang, respectively, used a 50-zhang-square grid that was further developed at Yuan Dadu and Ming-Qing Beijing, punctuating the use of different scales and modular proportions for different building sizes and complexes. (Fu, Steinhardt, and Harrer 2017, 329–361; Fu 2004, 319–355). JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 459 pffiffi √2and times in proportional treatment, which incor- Zhoubi Suanjing. Liu Hui comments on the first three porate the intimate associations with the circle-square problems of the chapter, which use the case of base and square-circle diagrams. His results show that the 3, altitude 4 and hypotenuse 5 to illustrate how each geometricratio canbeutilizedinplanningand designing side may be found using the other two. Specifically, cities and buildings through actually measured and Liu explores instructions for finding the square root restored drawings. (Wang 2017a, 2017b, 2017c, 2018a; of the hypotenuse from the sum of the squares of the Wang 2018b, 2019) other two sides (Cullen 2007, 88). Another research has explained the 1: √2 relationship In this overall perspective, this thesis intends to ferret for ancient building constructions in the Asuka Period. On out the proportional systems of the interior timber- thebasis of theratio 1: √2, the study proves the employ- frame structures at Pusŏksa Buddhist Monastery 浮石 ment of design methods, which define the distance from 寺, which is considered to be one of the oldest buildings the pagoda foundation to the diagonal line, and its corre- in Korea. So far, no surviving wooden buildings con- lation to the depth of the halls, showing that Japanese structed during the Three Kingdoms period have been craftsmen made use of a right angle ruler. (Ono 1964, discovered in the Korean Peninsula. Among the build- 623 –, 625) Likewise, major tools for drawing and mea- ings left, the KŭngnakchŏnHall 極樂殿 of the surement are depicted in surviving examples of illustrated Pongjŏngsa Buddhist Monastery 鳳停寺 is believed to religious materials such as the images of Fuxi and Nüwa be the oldest in Korea. in the Wuliangci bas-relief or Astana Tomb, which show them holding an L-square ruler and a compass (Figure 2). Although the philosophical background and evidence of 1.4. Pusŏksa Monastery and its Muryangsujŏn the √2 application are still insufficient in building con- Hall struction terms, they offer a key mechanism to an under- The MuryangsujŏnHall無量壽殿 of Pusŏksa Monastery standing of ancient design rules in East Asia resulting 浮石寺 was initiated by Ŭisang 義湘 (625–702), the from the √2 implementation as a key principle. purported founder of Hwaŏm in Korea. The Pusŏksa Monastery was not as large as it was at the time of its foundation, estimating that the monastic complex 1.3. Research methods and scope developed into a large-scale temple in the 9th century, This paper provides an in-depth discussion of the the late Unified Silla. In early Koryŏ times, it was said that square root of 2 , using mathematics and astronom- Wŏnyung guksa 圓融國師 (964–1053) preceptor ical books, such as the Jiuzhang Suanshu (The Nine reformed the Muryangsujŏn Hall in the 7th year of Chapters on the Mathematical Art) and the Zhoubi King Hyŏnjong (1016) (Munhwajaegwalliguk 1976). Suanjing (The Arithmetical Classic of the Gnomon According to the Pongwangsan pusŏksa gaeyŏn’gi 鳳凰 and the Circular Paths of Heaven), which was written 山浮石寺改椽記 (Repair Records at Monastery Pusŏksa by an anonymous author around the 1st century BC. in the Phoenix Mountain), since then, the Muryangsujŏn Korea’s ancient history book, the Samguk sagi Hall was burned by Japanese pirates during the 7th year (History of the Three Kingdoms), mentions in the of King Kongmin’s reign (1358) at the end of Koryŏ chapter seven of the Silla government offices in Dynasty. In the 21st year of King Kongmin (1372), Book 38 that the Jiuzhang Suanshu was used in the Wŏnŭng guksa 圓應國師 (1307–1382) preceptor was 7th-8th centuries as a regular mathematics textbook appointed as the Pusŏksa’s chief priest at the king’s in the Sanhak department 算學科 (arithmetic educa- command, and in the second year of King U (1376) tion) at the Silla-era Kukhak 國學 (the state-run edu- during Hongwu 9, he repaired the Muryangsujŏn Hall. cational institution). It is possible that the Jiuzhang These evidences were confirmed by explanatory Suanshu entered the peninsula at this time. The Nine legends uncovered in 1916 during the dismantlement Chapters of the Jiuzhang Suanshu examine Gougu 勾 and repair works of the MuryangsujŏnHall. (Cultural 股 (base and altitude), in close association with the Heritage Administration of Korea 2002,63–65) The square root of 2 is an irrational number. It cannot strictly be measured with a ruler based on our unit of measurement, no matter how small we mark fractional subdivisions. However, when we calculate the length of the diagonal of a rectangle, employing the Pythagorean Theorem, we obtain, indirectly, an irrational number (Carnap 1995). Two records on the Muryangsujŏn Hall’s repair works in 1016 and 1376 were found within the corner bracket set at the northwest, and on wooden members at the southwest corner toward the front facade. The former inscription is written as follows: 此寺唐高宗二十八年儀鳳元年新羅王 命義相法師始立創建 後元順帝十七年 至正戊戌敵兵火其堂 尊容頭面 飛出烟焰中 在于金堂西隅 文藏石上而奏于上泊洪武九年丙辰圓融國師改造改金而至于萬曆三十九年辛亥五月晦日風雨大作柝其中樑明年□壬子改椽新其畵彩 儼若旧制也 記其匠碩及勸緣人以示後也. The latter is written as follows: 此金堂自洪武九年經倭火後改造而至萬曆三十九年自折衝椽也壬子年始役畢於癸丑年八月也. They record that the building was reformed by Wŏnyung圓融 guksa preceptor in 1376. However, it is believed that the chief monk of the Pusŏksa was originally Wŏnŭng圓應 guksa preceptor at that time. To put it exactly, the name Wŏnyung was a wrong record, and all inscriptions on repair works drawn up in 1612 ~ 1613 (in the 40th to 41st year of the Wanli era, i.e. the 4th to 5th year of King Kwanghae) present a time lag of about 240 years from 1372. (Munhwagongbobu Munhwajaegwalliguk 1980,16). 460 J. CHA AND Y. J. KIM Figure 2. Painting of Fuxi and Nüwa holding a compass and an L-square ruler, 7th century Astana Tomb in Turpan, Xinjiang Uygur, China. Source: National Museum of Korea. JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 461 Great attention has been shown to the question pillar tops along both building axes; the second, on regarding the range of the 1376 repair work. Not only closer inspection, the MuryangsujŏnHall’sstone is the inscriptions recorded about 240 years later, but foundation platform with subsumed footings for both the extent of the fire loss and the magnitude of round columns follows the construction method in the reform are obscure in that the two records empha- the Unified Silla period since it bears a resemblance size ‘recomposing a building and repainting a statue to the developmental stages that imitate the outline with gold simultaneously 改造改金’ and “remodeling of timber frameworks in performing masonry work a building 改造,” rather than its reconstruction. (Kim 2011, 306, 447). Such being the case, there have been many obser- During the Japanese forced occupation of Korea, vations that the architectural forms and techniques on moreover, Sekino Tadashi identified the Pusŏksa the Muryangsujŏn Hall seem to utilize much older Monastery’sMuryangsujŏnHallas the oldest building methods than those shown in the 14th century’s build- on the Korean Peninsula. In 1911, Sekino wrote that ings (Han’gukpulgyoyŏn’guwŏn 1988, 88). In compar- he first discovered wooden architecture built in the ison with the architectural features of the Chosadang Koryŏ era in an article. He made a brief report about Hall situated in the same monastic site, the academic the location and history of Pusŏksa Temple, along generalized perspectives guess that the Muryangsujŏn with its construction age and architectural value of Hall was constructed 100 to 150 years earlier (Sekino the Muryangsujŏn and Chosadang Halls (Sekino 1941, 733; Munhwagongbobu Munhwajaegwalliguk 1913). He praised the two buildings, commenting 1980, 150; Chŏng 1974,27–32). “the Muryangsujŏn and Chosadang Halls are indeed Likewise, there are interesting researches in com- very artistic and skilful buildings in the Koryŏ period, parison to other extant examples in the Koryŏ age and the traditional multi-coloured paintwork of the such as the Monastery Pongjŏngsa’sKŭngnakchŏn buildings was truly ecstatic. I (Sekino) believe that Hall, the Monastery Sudŏksa’sTaeungjŏn Hall, and they are enough not only to be the oldest known ancient Chinese buildings, Han argues that the mod- buildings discovered but also to regard it as the great- ification in 1376 was about repairing some of the lost est exquisite beauty among wooden buildings in the parts at the time of the fire, and there are claims that Koryŏ times.” Sekino evaluated the architectural value the existing Muryangsujŏn Hall retains its original of the Pusŏksa Muryangsujŏn Hall, explicating “it is status as it was when the preceptor Wŏnyungguksa an excellent building analogous to the masterpieces reconstructed the building in 1016 (Han 2002, 148). of the Kamakura period. The construction style is Cha’sresearchpointsout that theMuryangsujŏn particularly commensurate with ancient Japanese Hall’s diagonal beams joined with corner bracket architecture. The internal framework built above clusters are similar to the construction method of bracket-sets and the underside of the tiled roof pre-11th century wooden buildings in northern bears a striking likeness to the Tenjiku (Daibutsu) China, as its universal applications are mainly seen style in the Kamakura era, and the shuttle-shaped through the examination of edifices built in the Tang columns with entasis are similar to those in the build- (Foguangsi Monastery’s Todaidian Hall, built 857), ings at Horyuji Monastery.” (Sekino 1913,3-5) the Song (Yongshousi Temple’s Yuhuagong Hall Likewise, according to the field cards recorded by (destroyed), built 1008; the Huayansi Monastery’s Sekino in 1913, one year after the investigation of Bhagavat Sutra Repository built 1038), and the Liao Pusŏksa Temple, he found the ŬngjinjŏnHallat Dynasties (Monastery Dulesi’s Guanyinge Pavilion, Sŏgwangsa Monastery, another wooden building in built 984), and gradually degenerated after the Jin the Koryŏ era, in Kangwŏn-do Province (Sekino 1941, Dynasty (Cha 2014, 131–142). Kim Dogyŏng notices 645–650). Sekino recognized the above-mentioned the problem that wooden members which consist of three buildings would be constructed in the Koryŏ the column-top bracket clusters and column-top tie era (Sekino 1941, 645–650, 723–742). After that, apro- beams to make a timber-framed structure keeps the pos the construction era and building characteristics same size with regular measurement, asserting the of the Muryangsujŏn and Chosadang Halls, On remarkable assumption that the MuryangsujŏnHall 27 October 1923, in the Chōsenshi gakkai 朝鮮史學 might be partly repaired in the 14th century, cher- 會 (Chosŏn History Society), he announced a paper on ishing its original shape in the 11th century (Kim the subject of “Chōsen saiko no mokuzō kenchiku 2014, 152). Kim Youngjae recognizes that the [Korea’soldestwoodenbuilding]” Histhesisnoticed Muryangsujŏn Hall upholds old construction meth- the difference between the architectural styles of the ods in two regards: the first, the Muryangsujŏn Hall two buildings and concluded that the Pusŏksa does not serve the extension of architrave through Muryangsujŏn Hall was at least 100 to 150 years column top at the corner, while the extant buildings older than the Chosadang Hall 祖師堂 (a hall of the constructed during the Song and the Liao Dynasties founder). Sekino confirmed, in a manuscript exca- across China have architraves that pass through the vated during the repair work of the Chosadang Hall, 462 J. CHA AND Y. J. KIM that the Chosadang Hall was built in the third year noteworthy is that various ancient documents asso- (1377) of King U in the Koryŏ Dynasty and was ciated with mathematical ideas are presented. repaired in the 21st year (1490) of King Sŏngjong in Among them, concerning the application of the square the Chosŏn Dynasty, although he mentioned the root of 2 (√2), the Qujingwei 取徑圍 (geometrical tie- needs of additional comparative studies with other up between diameter and circumference) proposes extant Korean buildings in the late Koryŏ and the the fundamental rule about the approximate rate of early Chosŏn Dynasties. Sekino Tadashi asserted that the slanting length, “If one side of a square is 100 in the Muryangsujŏn Hall, which had no comparable length, its diagonal length is 141 as the numerical examples in Korea and had no contemporary account value. 方一百, 其斜一百四十有一,” following the of that time, was built in the early 13th century. numerical principles in the Jiuzhang Suanshu, criticiz- (Sekino 1923, 1941, 723–742) ing that the “square seven oblique ten方五斜七 (if one In the light of these considerations, the Muryangsujŏn side of a square is seven in length, the diagonal length Hall which will be discussed in this thesis, is strongly is ten as the numerical value)” reflects lots of negli- believed to have been built between the 11th and 13th gence (Li 1103 (Song), 22), although the design for- centuries in the fact that the MuryangsujŏnHallemploys mula “square seven oblique ten” in the combination of significant structural elements such as architraves without rational numbers is more precise and closer to the any extension through the columns at the corners, diag- integer ratio of √2, an irrational number. Such design onal beams merged with corner bracket sets, the repeti- tradition which had been long preserved by profes- tive utilization of regular size members, as well as the sional builders solves for easier building constructions acceptance of the pre-Koryŏ’s methods to establish the problem of √2 times unable to be strictly measured a stone foundation platform. with a ruler grounded on the unit of measurement. Hence, this research examines an arrangement of Likewise, certain contents of the Zhoubi Suanjing are foundation stones on a ground plan, together with an included as follows: “Shang Gao answered. Numbers interior timber-frame structure, with a focus on the and their law – arithmetic – derive from the circle and Muryangsujŏn Hall at the Pusŏksa Buddhist Temple. the square. The circle arises from the square, the square This thesis delves into fundamental design principles from the carpenter’s L-shaped try square, and the car- with proportional systems in that all columns inside penter’s try square from multiplying nine by nine and the building are well arranged at regular distance with- getting eighty-one. The myriad things, each meeting out their reductions and movements, keeping an old their roles, are measured by the circle and the square; construction method, and in that proportional regula- whereas the chief architect, to create models and styles, tion in the application of the square root of 2 (√2) for its has devised the pair of compasses and the carpenter’s construction are proportionate to those of Chinese try square. He impairs a square to make a circle or breaks buildings in the contemporary period. The following a circle to make a square. A circle that fits in a square is studies on the Koryŏ buildings should be more pro- called an inscribed circle in a square, while a square that duced as future research tasks. fits in a circle is called a circumscribed circle. This shows that the Yingzao Fashi considers the Gougu yuanfang tu 勾股圓方圖(Drawing for bases, altitudes, 2. The Yingzao Fashi, and the mathematics circles, and squares) in the Zhoubi Suanjing.(Figure 3) books in ancient China, the Jiuzhang Suanshu Furthermore, the dialogue of Shang Gao in the Zhoubi and the Zhoubi Suanjing Suanjing includes an in-depth discussion on the relation- ship between circles and squares, along with the concept Importantly, the imperial Northern Song dynasty (- of Tianyuan difang (Heaven is round and Earth is square). 960–1127 CE) treatise on architecture, the Yingzao The text reads, “The square belongs to the earth, and the Fashi 營造法式 (Treatise on Architectural Methods, circle belongs to the sky; the sky is round and the earth is 1103 CE), contains a series of numerical values and square.The number of thesquareisbasic,andthecircle terminology for constructing buildings. At its head, comes from the square (方屬地,圓屬天,天圓地方.方數 the Yingzao Fashi has a table of contents and the fore- 爲典, 以方出圓).” In accordance with Zhao Shuang’s word, “Kanxiang 看詳” (Examination of details). The commentary, this means that the earth is stationary treatise classifies all architectural work into thirteen while the sky revolves in motion, a cosmological concept systems recorded in thirteen chapters, including the universally shared by the ancients (Chen 1984, 95). The Damuzuo大木作 (structural carpentry and woodwork) sentence ends with, “So whoever knows the earth is wise, and Xiaomuzuo 小木作 (joinery and non-structural andwhoever knowsthe skyisholy. Wisdomcomesfrom carpentry). In the “Kanxiang,” what is unusually An article with the same subject as the lecture was published in the same year. Sekino Tadashi, 1923. “Chōsen saiko no mokuzō kenchiku [Korean oldest wooden building],” Chōsen to kenchiku [Chosŏn and Architecture], Vol 2, No. 8, Chosŏn Architecture Association. The paper was also subsumed in a book, which was compiled in 1941. Sekino Tadashi, 1941, Chōsen no kenchiku to geijutsu [Korean Architecture and Art], Tokyo: Iwanami Shoten. 昔者周公問於商高曰:數安從出？商高曰:數之法出於圓方.圓出於方.方出於矩,矩出於九九八十一.萬物週事而圓方用焉,大匠造製而規矩 設焉. 或毀方而為圓, 或破圓而為方. 方中為圓者謂之圓方；園中為方者謂之方圓也. (Li 1103 (Song), 19–21). JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 463 Figure 3. Line drawings of Gougutu by Shang Gao (revision version) and Fangyuan Yuanfangtu [Rounded-Square, Squared-circle Map] by Shang Gao (restoration version) (Cheng and Wen 2012). the base, and the base comes from the bend. It controls the rectangle (ju) refers to an L-shaped try square making everything through digitizing it by using a curved line (是 asquare – these represent the tools used to draw circles 故,知地者智,知天者聖 智出於句,句出於矩.夫矩之於 and squares individually. The weights and measures 數, 其裁制萬物, 惟所爲耳).” The above-mentioned con- (duliang), on the other hand, represent a standard unit cept (Figure 3), which considers squares and circles on of metrical type that calculates the length, width, depth, a mathematical basis, is similar to the description of the volume, and capacity of myriad things in the world, and Gougu yuanfang tu (Drawing for bases, altitudes, circles, reflects numerical rapports between innumerable things. and squares) in the Yingzao Fashi of the Song dynasty. It In other words, standards and measurements always illustrates how combining squares and circles can control represent the formats of space and quantitative ties. In myriad things, which can be converted into various things ancient times, in China, geometry was associated with (Cheng and Wen 2012, 12). Likewise, the Jiuzhang number and amount, and all matters were solved through Suanshu, another ancient mathematics book, explicates arithmetic and algebra (Guo 2009,7). Itsintroductionalso the L-shaped try square and compass as measurement indicates that a change in worldview expounds the cir- devices that can embody innumerable things (Shen et al. cumstances of myriad things. “Fu Xi painted the eight 1999, 520-522). In line with the preface commented by Liu trigrams (bagua) in remote antiquity to communicate Hui in the 3rd century, the book evolved from the Jiu shu the virtues of the gods and parallel the trend of events 九數 (Nine Operations with Numbers), which dates from in the earthly matter, and then he invented the nine-nines the time of the Duke of Zhou in the 11th century BC. It algorithm (jiujiu) to co-ordinate the variations in the hex- describes a mathematical method similar to that of the agrams (liuyao)(昔在庖犧氏始畫八卦,以通神明之德,以 Zhoubi Suanjing: “Even those results delivered by the law 類萬物之情, 作九九之術, 以合六爻之變).” Led by the handed down through the generations should be pre- virtues of the gods, mathematics became a symbolic lan- sented by measuring the length with a compass and an guage, its main function being to classify the state of all L-shaped try square (至於以法相傳,亦猶規矩度量,可得 things (Guo 2009, 2). In this way, the key idea in both 而共).” The law (fa) sets out mathematical principles, the books is that myriad things in neighbouring areas can be regulation (gui) refers to a compass drawing a circle, and created using circles and squares. The diagram of an 464 J. CHA AND Y. J. KIM inscribed circle and a circumscribed circle in Figure 2 researchers believe that the interior timber-frame struc- shows that circles and squares are the basic components tures were much transformed by the repair work (Han for construction. 2002,139–48). In the main hall, the structural system of Furthermore, Liu Hui states that the Yellow Emperor a diagonal beam in the corner of buildings, which transforms and extends the cosmic principle of the included a bracket complex and column, is tantamount trigrams tremendously to solve practical problems, tothesideframestructuretechniquesoftheTangandthe such as divination, regulation of the calendar, and Liao periods. The hall has attracted attention due to harmonization of the musical scale. The prefatory research that identified construction techniques from chapter “Kanxiang” of the Yingzao Fashi, as has been before the 12th century, which correspond to northern noted earlier, mentions various phrases that relate to East Asian architecture in the contemporary period (Cha the length of figures and buildings. The parts of rec- 2014, 131–142, 2016,78–103). But, the MuyangsujŏnHall tangles are as follows: “If the lateral side of a square is experienced a typical design process in the fusion with one hundred in length, its diagonal line across the the southern and the northern architecture of China square is one hundred and forty-one in length (方一 (Kim 2011, 538–540). It is a five-by-three–bay building 百其斜一百四十有一).” In this passage, the square with a front façade and side façade measuring 18,751 root of 2 (√2), or the length of a diagonal line across meters and 11,511 meters, respectively. There are no a square, is notable. The simplest mathematical form of inner columns that have been migrated or removed. In √2 is an equilateral triangle. Assuming that the equi- particular, both the building platform and foundation lateral triangle’s base and height measure 1 at right stones that are presently visible (the natural interior foun- angles, the length of the hypotenuse of the triangle is dation stones are elements that were replaced during the defined as the square root of 2 or 1.414 when Japanese forced occupation) are a variation that is rarely expressed as a numerical value in modern mathe- seen after the Unified Silla period. Some of the original matics. A square results from combining two equilat- styles are inherited by recycling elements previously eral triangles. The method prescribed in the Yingzao used in the Koryŏ period at the time of its construction. Fashi to determine the length of figures is based on the (Figures 4 and 5) selection of the Pythagorean Theorem. The formative On the ground plan of the Muryangsujŏn Hall at ideas of the ancients, in consort with the composition Pusŏksa Monastery, the central front facade bay and principle of circles and squares, which are embedded the bays on either side of the central bay along the in the Zhoubi Suanjing and the Jiuzhang Suanshu, front facade are the same widths, while the outermost permeate the contents of the Yingzao Fashi. bay is narrower than the other three. (Figure 6)There are some exemplary buildings in East Asian architecture in which the ratio of the central bay to the side bays is 3. The proportional system of timber-frame 1:1. To put it briefly, the 7th century main hall at structures in the Muryangsujŏn Hall at Daikandaiji Monastery (Nara), the late 8th century main Pusŏksa Monastery hall at Kamŭnsa Monastery (Kyŏngju, North Kyŏngsang), the 10th century main halls of Buddhist Monasteries in Pusŏksa, located in Yŏngju City, North Kyŏngsang Parhae, the early 11th century main hall at Fenguosi Province, is a Buddhist monastery established by the Monastery (Yixian, Liaoning), and so on (Kim 2011, monk, Ŭisang, and one of ten monasteries grounded on 304–312). The side facade of the building is a timber Hwaeom (Avatamsaka) thought. The Muryangsujŏn(the framework with two smaller bays on either side of its Hall of Immeasurable Life) was constructed as the main central bay. The inside of the building is therefore rela- hall with a hip-and-gable roof in the chusimp’o style, tively wide. The interior stone plinths and columns are following bracket complexes placed only at the heads of arranged on the same line as the external elements. the building’s structural columns. As aforementioned, the Previous studies have generally had a strong tendency building was completely dismantled and then reas- to regard a central bay as an absolute standard for sembled for the repair work in 1916 during the 10 framing a wooden building, but this paper offers Japanese forced occupation of Korea . Korean Liu Hui describes the ratio of diagonal length in the octagonal geometrical construction method given in the Jiuzhang Suanshu, and argues that the error of the square root of 2 (√2) is approximately 1% (Liu 2014). The repair works of the Chosadang and the Muryangsujŏn Halls were performed during four years from 1916 to 1919 at a cost of 23,566 yen (Kim 2011,440). Kim maintains that the Pusŏksa’s Muyangsujŏn Hall is a typical building combined with directly opposed notions between the south and the north of Chinese architecture; in the south china the one is the acceptation of dingtougong (half-bracket arm) with a tenon-and-mortise to stabilize the combination of the longitudinal with the transverse beams and the columns of the building; in the north China the other is the employment of tuojiao (inclined struts) between one transverse beam and another transverse beam in comparison with the construction method of the Pongjŏngsa’s Kŭngnakchŏn Hall (Kim 2011, 539). The width of the central bay and the side bays are slightly different, but they are all about 4,200 mm long (Cultural Heritage Administration of Korea 2002, 12). Interesting suggestions for the usage of the central bay are offered by some researchers as both a functional space and a significant factor to make room for monks and laypersons. Kim and Pak note that, since the 7th century (the Unified Silla), as the function of the central bay becomes more important over time, the central bay tends to be wider than the outermost bay on the ground plan (Kim and Pak 2008,20–21). Kim Youngjae puts forward the very interesting hypothesis that the particular employment through the long width of bays might be intended to express the authority as a Buddha hall, and it might help adherents in a sense of awe contemplate the Buddha images at a long distance (Kim 2011, 309). Additionally, through the comparative JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 465 Figure 4. Front view of the Muryangsujŏn Hall at the Pusŏksa Monastery. Figure 5. Interior timber frame structures of the Muryangsujŏn Hall. a new interpretation in the form of an arithmetical columns, both an inscribed circle with a radius of one commentary, which is contrary to existing views. In the (1) and a circumscribed circle with a radius of the square MuryangsujŏnHall atPusŏksa Monastery, depending on root of 2 (√2) can be drawn on the ground plan. The the direction in which columns and eaves purlins are inscribed circle meets the second column to the left and placed, and assuming that the width of the outermost the corner edges of the building platform when looking bays is one (1) and that the location of the corner at the southern front facade, while the circumscribed columns is regarded as a central point to draw figures, circle matches the position of the interior corner col- the authors can draw squares or circles, and the lateral umns. All columns in the design of the building are length of a square can be considered as two (2). As seen arranged at intervals of √2 times, assuming the width in Figure 6, based on the center point of the corner of the outermost bay is one (1). In contrast to the analysis between extant middle-gates at ancient temple sites and historical remains, Han argues that the restoration plan for the middle-gate’s archaeological remain at Mirŭksa Monastery can be carried out looking upon the central bay on the ground plan as an important factor (Han 2012, 206). 466 J. CHA AND Y. J. KIM Figure 6. The ground plan of the Muryangsujŏn Hall (Cultural Heritage Administration of Korea 2002, 121). common building standards of the timber framework, is presumed to be fairly similar to the Tang cubit system this building demonstrates that the width of the outer- 唐尺 (tangch’ŏk), which is equivalent to 29.694 cm, or most bay is the basis of the column arrangement, not the length of the cha, a unit of measurement used dur- that of the central bay. On the contrary, despite a few ing the Unified Silla era. Yoneda Miyoji discovered the errors in the placement of roof purlins, on a closer look adoption of the Tang cubit (29.694 cm) by conducting an at the side façade of the building in the direction of the investigation of Sŏkkuram Buddhist Grotto and Pusŏksa transverse beams that form the roof, the distance Monastery. The measurement was different from the between the side and central bays at the side façade is length of the cha used at that time, which was identical tuned to a ratio of 1: √2 + 0.414, on the assumption that to a kokch’ŏk 曲尺 (30.303 cm), and implies the use of the distance between the eaves and the interior col- a carpenter’s try square. Measuring each stone element umns is one (1). This numerical value is understood as of the Buddhist temples, he noted that the same mea- an arithmetical concept rather than a geometric con- surements, equal to 0.98 kokch’ŏk,1.96 kokch’ŏk,and cept. Furthermore, the width ratio of the outermost bay 23.6 kokch’ŏk, were used repeatedly. He then concluded to the central bay or the side bay toward the front that the cha used by architects and stonemasons at the façade is 1: √2. This shows that the proportional system time of Unified Silla was equal to 0.98 kokch’ŏk,i.e. in the synthesis of 1 and √2 is applied to the interior 29.694 cm, and that it was a unit of measurement used ground plan. The outermost front façade bays are in the Tang dynasty. He named the cha the Tang cubit, 3,034 mm long, and the central bay is 4,219 mm long. the reference scale of the Tang Dynasty (Yoneda 1944, √2 times the length of the outermost bays is 4,290 mm. 1976,26–28). Units of length in Chinese measurements This shows a margin of error of about 1.7 percent from were rooted in human dimensions. These origins were the actual measured length of 4,219 mm (Cultural comparable to those of Greek metrology; however, the Heritage Administration of Korea 2002, 128). In addition Greek and Romans preferred the foot as a unit of mea- to the arrangement of the columns on the ground surement, while the Chinese, Korean, Egyptians, Ancient plan, by looking carefully at the tie-up between the Indians, and Mesopotamians preferred the cubit based building platform and columns, it can be seen that the on the forearm length from the tip of the middle finger distance from the corner edge of the building platform to the bottom of the elbow. The other Unified Silla to the corner columns is almost the same as the width of constructions built at the sites of the Mangdŏksa the outermost bay. The ratio of the distance between Monastery, Sach’ŏnwangsa Monastery, and anonymous the corner edges of the building platform and the cor- Buddhist site in Ch’ŏn’gulli District employed the Tang ner columns to the distance between the corner eaves cubit (Kim 2007, 456). Likewise, in the Isŏng sansŏng columns and the interior corner columns is 1: √2. Mountain Fortress, Hanam City, Kyŏnggi Province, As seen in Figure 6, it is highly probable that the ratio a wooden ruler estimated to have been used in the of 1 to √2 is applied to define a ground plan that situates Silla era before and after the unification of the Three columns, and the width and depth of the building foun- Kingdoms in the 7th century was excavated in 1999 dation are planned in the same way. This proportion (Chungang ilbo 1999, December 21). The ruler, 29.8 cm JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 467 in total length, had nine graduations engraved on the the other figures cannot be confirmed in the current side with equal spacing, which followed an identical line drawings. However, the most remarkable thing measurement system to the Tang standard ruler, and is that both the √2 times the length between the has been universally used in East Asian countries since its central bay and the outermost bays along the front adoption during the Tang dynasty. It is interesting to façade of the ground plan and the ratio of actual note that the numerical value of 21 times is produced height of the eaves columns to the height of the without error when 29.694 cm based on the Tang cubit is eaves purlins share a margin of error less than divided by √2 (= 1.414). Thus, the Tang cubit and √2 two percent in comparison with the actual measure- make a very good pair and merit future study, given that ments (Cultural Heritage Administration of Korea a Tang cubit ruler in the Kyŏngju capital during the 2002, 133). Unified Silla period has yet to be discovered. Interesting results are in the fact that the A cross-section of the MuryangsujŏnHallshows MuryangsujŏnHallisabuilding made of wooden thenumerical valueof √2 plus the concept of an components with shrinkage and expansion due to arithmetical and geometric sequence more clearly. seasonal temperature change in process of time, It can be seen that the basic ground for designing together with the maintenance records of dismantle- the building is the height of the front and back eaves ment and repair works. Besides, the criteria are the columns with column-top bracketing, and the height vertical distance from the upper side of the building of the eaves purlins as well. On the premise that the platform to the lower side of the eaves purlins, height of the eaves columns is one (1), the height of located between intermediary bracket-like timbers the eaves purlins is √2 times. To put it plainly, the (between the lead bracket arm (linggong)and the authors can draw a square corresponding to the purlin) and eaves rafters. In tune with that of the height of the eaves columns, as well as an inscribed Muryangsujŏn Hall, as seen through Wang’s research circle and circumscribed circle along the corre- (Wang 2011a, 2011b), Chinese wooden architecture sponding vertices of the square. The eaves purlins applies √2 times the vertical distance from the upper then meet at the point where the radius of the side of the building platform to the upper side of the circumscribed circle becomes √2times.From eaves purlins. This shows that there is a definitive a practical view through measured values, the height difference between the proportional standards of of the eaves columns is 3,460 mm, and the vertical Korean and Chinese architecture, although the rule length from the upper side of the building platform of √2 times is equivalently applied at a regular dis- to the lower side of the eaves purlins is 4,846 mm. √2 tance. These distinctions appear differently, drawing times the height of the eaves columns in cross- on country, region, and period. (Figure 7) section is 4,892 mm. This represents an error of Additionally, the height of the eaves columns is approximately 1% at the distance of 4,846mm to a standard distance when compared to various internal the lower side of the eaves purlins. It is judged that elements. The height of the interior columns is √2times Figure 7. Cross-section of the Muryangsujŏn Hall (Cultural Heritage Administration of Korea 2002, 154). The Tang cubit ruler, 29.694cm, which was used during the Unified Silla period, is a generalized theory widely used among Korean scholars. In addition, according to Zhang Shiqing, a Chinese scholar, he confirms that there are twenty-six examples of Tang cubit ruler and Tang cubit ruler at Shoshoin正倉院 in Japan, most of which are measured between 29.5 ~ 29.7cm in length. (Zhang 2004, 77) Therefore, it can be seen that the Tang cubit ruler used in the Korean Peninsula was ratified to 29.694cm, which is intimately associated with 21 times the numerical value of √2. In order to verify the correlation with the ratio of 21 times and the square root of 2, more detailed researches are needed in the future. 468 J. CHA AND Y. J. KIM when compared with that of the eaves columns. Double distributed, the distance between each purlin center the height of the eaves columns is equal to the distance could be different due to the different tilt of each purlin; from the interior column bases to the middle roof purlin. however, the distance between the centres of each The elevation of the middle roof purlin measures twice purlin is usually the same, excluding the eaves purlin. the height of the lower eaves column. The proportional A feature seen more clearly on the cross-section than on interdependence between the middle roof purlin and the the ground plan is the fact that the distance from the eaves columns is currently revealed not only in the two front interior column to the back (the distance from the Tang buildings (the Main Hall of Nanchansi Monastery front high column to the back inside the hall) is equiva- and the East Hall of Foguangsi Monastery at Mount lent to the distance from the interior high column to the Wutai), but also in Liao and Song buildings, including side border of the building foundation. The height of the late 10th century Guanyinge Pavilion of Dulesi the eaves columns is closely related to the length of Monastery, the early 11th century main hall of Baoguosi each interior element, and the intimate relevance can be Monastery (Ningbo, Zhejiang), the mid-11th century main found in several parts. When multiplying the height of hall of Shanhuasi Monastery (Datong, Shanxi), and so on the eaves columns by √2, it matches the position of the (Fu, Steinhardt, and Harrer 2017, 209–214; Fu 1998, eaves purlin. The height of the eaves columns is the 147–167). The application of this principle appears in same distance from the center of the columns purlin to Unified Silla construction. (Figure 8)The Sŏkkuram the center of the middle roof purlin. It is also equal to the Grotto’s vertical scheme shows that the carvings of the distance from the outline surface of the interior columns eight Bodhisattvas on the shrine enclosure, including to the eaves columns. In addition, half of the height their pedestals, stand 12 cha high. The height from the from the interior column bases to the middle roof purlin carvings to the shrine enclosure’sceiling is12 timesthe meets the outline surface on the exterior columns. What square root of 2. (Yoneda 1944, 1976,27–28, 140–141) is remarkable is not the distance to the eaves column (Figure 9) center, but the distance to the outline surface of the The height of the eaves columns is also relevant to eaves column. To put it another way, if √2times the the depth between each purlin. The depth of a building height of the eaves column is multiplied √2 times, then is usually measured by the number of rafter lengths the height of the eaves columns is doubled, and corre- necessary to cover the building from front eaves col- sponds to the location of the middle roof purlin. This is umns to rear eaves columns, and is equivalent to the a demonstration of the concept of geometric progres- spaces between the purlins in cross-section. The dis- sion. Multiplying this doubled height by is equal to tances from the center of the middle roof purlin to the the distance from the eaves columns to the interior center of the ridge purlin, and from the center of the columns. The height of the eaves pillars is equal to the middle roof purlin to the center of the eaves purlin are distance from the center of the ridge purlin to the center the same as the height of the eaves column. Although of the middle roof purlin, and the distance from the the distance between each purlin is uniformly center of the middle roof purlin to the columns purlin. Figure 8. Cross-section of the Dongdadian (the Great East Hall) at Foguangsi Monastery (Fu 1998, 153). In the case of the main hall of Nanchansi Monastery, the eaves columns are about 3.86m high, while the vertical distance between the eaves column top and the ridge purlin is about 3.81m high. Since this hall is only four rafters deep, the ridge purlin corresponds to the middle roof purlin in cross-section. A similar example can be seen in the main gate of Dulesi Monastery. JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 469 Figure 9. Systematic proportion system in the vertical scheme of Sŏkkuram Grotto (Yoneda 1944, 1976, 140–141). It can also be applied to the distance from the columns √2 times the height ratio obtained from the purlin to the lower edge of the roof eaves. From the MuryangsujŏnHallatPusŏksa Monastery, there cross-sectional profile, the distance from the eaves col- should be numerous references to aid in their recon- umn bases to the interior column tops is similar to the struction on the ruined monastic sites of the Kyŏngju distance from the eaves column bases to the ends of the Historic Areas, which are supposed to be constructed flying rafters. Moreover, when the height of the eaves with the Tang cubit of the Unified Silla period. columns is multiplied √2 times, it is equal to the distance However, since the height of the column cannot be from the eaves column tops to the interior column confirmed at the ruined monastic sites, the method bases. Applying √2 times or twice the height of the for estimating the height of the columns by judging eaves columns reveals an intimate connection with the the width of the stone plinths and the distance distance from the one column to the other, from between each plinth on the center should be studied the column tops to the building foundation, and from in the future. The fact that the concept of √2numer- the column bases to the ends of the eaves purlins. This is ical value is found in connection with wooden ele- to say that the measurement system is based on the ments from the MuryangsujŏnHall, whichis Tang cubit, which was widely used during the Unified recognized as a Koryŏ building in Korean Peninsula, Silla period. In both the ground plan and the cross- should be further complemented by an effort to com- section, √2 times and twice the height or distance of pare it with such coincident and related fields of study the eaves columns, depending on the distance between as wood-frame construction in ancient East Asian each column and the distance between each purlin, is civilization. applied as a regular proportional concept in construct- ing a building. 4. Discussion Thus, the proportional concept of √2times and two times can be seen as an important proportional By dwelling upon the philosophical background of concept in the construction of interior frame struc- ancient mathematics references in ancient China, the tures for wooden architecture. Through the proof of Zhoubi Suanjing and Jiuzhang Suanshu, this paper 470 J. CHA AND Y. J. KIM analyses the proportional system of interior timber- diagonal beams combined with the bracket sets above frame structures, with the following results. First, the columns at the four corners of the MuryangsujŏnHall from among the contents of the mathematics books can be meaningful shreds of evidence that the Buddhist in ancient China, this paper shows that it is possible to monument performed a repair work in the 14th century represent all things as circles and rectangles, and to preserving the 11th century’s original appearance recon- replace circles and rectangles with each other, in con- structed after the fire,soitisexpected thatits specific nection with the ancient ideation, Tianyuan difang construction era will be further reconsidered through (Heaven is round and Earth is square), a philosophical future research efforts by comparison with other extant background in architecture and urban planning that timber-frame buildings across East Asia. was very deeply reflected in formative thought. Second, the proportional system of inscribed Disclosure statement and circumscribed circles made by circles and squares on a ground plan has a very close link with the ratio of No potential conflict of interest was reported by the authors. 1 and √2 (1.414) in shaping a wooden framed structure. These numerical values can be important criteria for References the composition of the square ground plan and the cross-section. Third, the ratio of the height of the eaves Carnap, R. 1995. An Introduction to the Philosophy of Science. columns to the height of the eaves purlins at the New York: Dover Publications. Muryangsujŏn Hall is 1: √2, and similarly, that of the Cha, J. 2014. “Chungguk Myŏngdae IjŏnKŏnch’ukkwa Pusŏksa MuryangsujŏnChŏn’gakpugujo T’ŭkchinge Kwanhan height of the eaves columns to the interior columns is Yŏn’gu [A Study on the Structural Characteristics between 1: √2. 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Journal
Journal of Asian Architecture and Building Engineering
– Taylor & Francis
Published: Sep 3, 2019
Keywords: Muryangsujŏn Hall at Pusŏksa Buddhist Monastery; √2; Zhoubi Suanjing; Jiuzhang Suanshu; Yingzao Fashi
