Traditional Steelyard-making in China and Mechanical and Mathematical Tradition Related

Jürgen RENN, Baichun ZHANG, Matthias SCHEMMEL


     Lever principle is one of basic principles of mechanics. If we try to have a deep understanding of discovery of lever principle, we may make a case study of traditional steelyard-making and historical records in China. Helped by the Chinese Academy of Sciences and the Chinese Research Institute of Metrology, Professor Jürgen Renn, Director of Max-Planck-Institut für Wissenschaftsgeschichte, and his assistant, Mr. Matthias Schemmel, visited China, and made an investigation of steelyard-making in the Tongzhou District (originally called Tongxian) of Beijing and Changsha of Hunan Province in August, 1998.[1] They consulted steelyard-makers on a series of questions, and then the makers answered them and showed how to make a steelyard. Prof. Renn and Mr. Schemmel recorded what makers said, and videotaped what makers did. Additionally, they took pictures. A few months later, Prof. Zhang Baichun summed up findings materials Prof. Renn and Mr. Schemmel had collected, and combined them with historical records of steelyards. After these three scholars had exchanged their viewpoints, this research report on came into being.

     Perception about lever was accumulated during the extensive application of technology in ancient China, and rules of practices were gradually formed. However, meaning of concepts, the relations between them, and the distance between the practices and theory of lever principle, are still should be clarified.

One. Brief Historical Retrospect

    ‘Quan-heng’ (权衡) is the term used to name all apparatuses for measurement of weight in ancient China, i.e. balances and steelyard (杆秤). During the pre-Qin days (i.e. before 221 B.C.), Chinese people had already started using ‘quan-heng’ and other apparatuses to weigh. The Qiu-bo-jun Bronze Weight (右伯君铜权), which is now kept in Museum of Chinese History, has been attributed to be an article from state Qin in later time of Spring and Autumn Period.[2] It has a ‘bi-niu’ (鼻纽, a hole on the top for threading a cord), which is not substantial to say that the subject is a moving weight(秤砣, ‘cheng-tuo’)of steelyard. During the Warring States Period, application of ‘quan-heng’ had become more popular. An intact ‘quan-heng’, an equal-armed balance, used during the transition stage of Spring and Autumn Period to the Warring States Period (about 4 B.C.-3 B.C.) was excavated at Zuo-jia-gong Mount, Changsha, Hunan Province.[3] ‘Quan’ (权) and ‘heng’ (衡) were frequently mentioned in early documents of the pre-Qin days. The Mohist gives a qualitative account to ‘quan-heng’ and issue of balance.

     Promoting the standard for measuring apparatus, which was made up by Mr. Shangyang and had been in practice for over a hundred years in the State of Qin, Qin-shi-huang (the first emperor of Qin Dynasty) standardized apparatus for measurement over the nation.[4] The Han Dynasty took over Qin’s practice. Lu-li-zhi (Record of laws and rules) in Han Shu, (Book on history of Han Dynasty) explains: “‘quan-heng’, i.e. ‘heng’ is balance; ‘quan’ means heaviness. ‘Heng’ may be achieved by balancing weight of an object with heaviness of ‘quan’.” And also: “‘heng’ merges while heaviness of ‘quan’ is equal to the weight of weighted object.”[5] This rule is applicable to balance at least. Archeology study indicates ‘quan-heng’ of Qin and Western Han dynasties were equal-armed balance, ‘quan’ was weights; after Western Han, some ‘quan’ became moving weight of unequal-armed balance.[6]

     Documents of Han Dynasty and later times imply steelyard had been in use in Eastern Han Dynasty. Zhongni-dizi li-zhuan (Profile of Students of Confucius) in Shiji (Records of the Historian) accounts: “a balanced situation of one-thousand-jun object would be broken if putting on minute weight on either side”(千钧之重,加铢两而移, ‘qian-jun zhi zhong jia zhu liang er yi’).[7] ‘Yi’(移) here means location of moving weight or the heaviness is going to change. It is talking about a steelyard if the moved object is ‘quan’ along the beam. Mr. Zheng Xuan (A.D.127-200) of Eastern Han Dynasty, annotated Yue Ling in Li Ji (The Book of Rites) as “beam of steelyard is named ‘heng’”, “‘cheng chui’ (moving weight) is named ‘quan’” (“秤上曰衡,“秤锤曰权”).[8] ‘cheng chui’ is supposed to be moving weight.

     There appeared official steelyards during North Wei period (A.D.386-534), even ‘cheng-lou’ (秤漏, steelyard clepsydra), a kind of chronometer too.[9] During later period of Tang Dynasty (A.D.618-907), one ‘liang’(两), a weight unit, was made up of ten ‘qian’(钱). In Song Dynasty (A.D.960-1279), one ‘qian’ was divided into ten ‘fen’(分), and one ‘fen’ consisted of ten ‘li’(厘).[10] Liu Chenggui, an official of Song Dynasty, made a precision steelyard of smaller size—‘deng-zi’(戥子). Its popular name is ‘deng-cheng’ (戥秤). Liu also cast a set of weight as standard during the reign of emperor Jingde, setting precise standards for moving weight, weight of the pan, length of beam of steelyard as well as extent of division.[11] As a standard of measurement, it consisted of two series: one followed Qin’s and Han’s non-decimal-based system, which was using ‘shu’(黍), ‘lei’(累), ‘zhu’(铢), ‘liang’(两) as measuring units; the other one was of decimal system which had been put into use after Tang Dynasty with ‘hao’(毫), ‘li’(厘), ‘fen’(分), and ‘qian’(钱) as units, whose maximum weighed capacity was one and half ‘qian’ and minimum measuring unit was ‘li’.[12] During Yuan Dynasty (1206-1368), the official steelyard gradually trended towards measuring apparatus with fixed standard for moving weight and steelyard. However, moving weight of that time hadn’t yet met requirement of precision.[13]

     In a book completed in 1638, Alvaro Semedo (1585-1658), a Portugal Jesuit, accounted: Chinese never weigh things with balance, instead, they measure lighter objects using steelyards. Beam of steelyard is made of wood but not iron; silver and copper dots on steelyard’s beam are used as indicator of scale. Chinese people have different kinds of steelyards. Steelyard of medium size has three lines of dots on beam, the first line is for indicating weight ranging from 3-5O.Z; the second line for heavier weight, climax of this scale is 10 O.Z; the last line is for 20 O.Z.. While weighing precious metals, such as gold and silver, or herbal medicine, white bony beams with black strokes as scale are used.[14]

     During Ming Dynasty (1368-1644) and Qing Dynasty (1616-1911), the Board of Works (工部) administrated making of measuring apparatus. During the first days of Qing, weighed capacity of steelyards of different sizes varied. Capacity of larger steelyards was a hundred ‘jin’(斤), and smaller ones were used to indicate weight ranging from 10 to 50 ‘jin’; steelyard with smaller pan may handle weight from 2 ‘jin’ to 16 ‘jin’, while larger ‘deng-zi’ was designed for weight ranging from 50 ‘liang’ to hundred ‘liang’, and smaller ‘deng-zi’ for 10 ‘liang’ to 30 ‘liang’.[15] The Board of Works ordered craftsmen to make steelyards and ‘deng-zi’ (small steelyards) while needed.

     After establishment of the Republic of China, the Ministry of Industry and Commerce took charge of standardizing apparatus for measurement. In “Bill on Standard for Measurement of the Republic of China” issued in July, 1928, it was stipulated: one third of a meter is one ‘chi’(尺), half of a kilo equals to one ‘jin’, one ‘jin’ is made up of 16 ‘liang’ (one ‘jin’ was 10 ‘liang’ in the original draft).[16] The government of Republic of China published “Law Concerning Measuring Apparatus of Republic of China” in February, 1929, in 1931 revised “Regulation on Implement” which was concerned with making, inspecting as well as testing of measuring apparatus. All stipulations concerning steelyard may well be summed up as followed:[17]

1. Hard-wearing and non-contractile materials are stipulated for measuring apparatus making; wood used is limited to the dried ones, while reactive metal should be coated with paint ;

2. Blades of measuring apparatus and parts contacted with the blades should be made hard and smooth, concerned materials are limited to steel, glass and jade. Parts of fulcrum and hanging-pan point (重点,‘zhong-dian’, on which a pan or a hook is hung) are supposed to be made of metals of proper rigidity;

3. Materials such as leather thread and flaxen thread should be applied in parts of fulcrum and hanging-pan point, if maximum weighed capacity of steelyard, whose sensitivity is not supposed to exceed one percent of weighting capacity, is below 30 ‘jin’;

4. Maximum number of the lifting cord of steelyard (秤纽, ‘cheng-niu’)is two. The two lifting cords should be set on either side of steelyard’s beam separately, hook or pan of steelyard are supposed of versatile function; this stipulation is not applicable to steelyard whose weighting capacity is below 30 ‘jin’;

5. Moving weights may be sorted into different categories according to their shapes, such as cone or pyramid. Iron moving weight should be properly cast with a hole, so that it’s easily to testify quality of metal used as well as to increase and decrease the weight of moving weight, weight of moving weight ones mustn’t lighter than one thirtieth of maximum weighting capacity;

6. Weight indicated by a scale interval may not lighter than sensitivity of a steelyard, the sensitivity should be below one tow-hundredth of weighed capacity;

7. While testing steelyard, followed procedures should be taken: first check weighed capacity; secondary, put on or take off a weight of sensitivity or the minimum weight that may be indicated by the shortest scale interval, see if the extent, that the beam moves, is more than one thirtieth of distance between fulcrum to end of the beam.

     Records about steelyard from documents of ancient China were quite rough, making it difficult for today’s people to learn complex details and knowledge about steelyard-making. Modern Chinese craftsmen’s practices of it provide plenty of information.

Two. Steelyard-making in the Tongzhou District of Beijing

    Demand for steelyards used to be quite large, all rooms at Weighing Apparatus-Overhauling Station of Tongzhou District were workshop for steelyard-making. Later, the state government promotes electronic system to replace steelyard, accordingly the output of it drops sharply. Mr. Ai Chun, head of the Measurement Bureau, kept two people continuing the production. Followed are the procedure of their practice of steelyard-making with weighting capacity of 15 kilo.

(One) Planing Beam

    Wood used to make steelyard is supposed of good quality, according to craftsmen. They pick up a certain kind of wood, make a bar into a beam for a try, check the beam every few days. After several rounds of test, it may be detected if the wood is of stale quality. They had tested imported woods, such as Litchi wood, but their qualities are unreliable. Besides hardwoods, such as mahogany and ormosia henryi, jujube wood is main material for steelyard-making in Beijing and northern area. Jujube wood probed to be of stable quality in tests and turns to be an ideal choice. Of course, the specific gravity of rods isn’t absolute same. Usually, specific gravity of wood near core of a tree is denser than that near skin, however, the difference wouldn’t be significant, but, when wood is wet, its specific gravity would increase.

     Xincheng, a county of Hebei Province, is a location famous for steelyard’s beam production. High-temperature heating and soda are used to treat crude material: i.e. put jujube wood into water to be seamed and boiled for an hour.[18] After two-month natural air-drying, next making phase may be started. At first a jujube rod is sew into bar carrying a square transversal section, then the bar is planed into a beam.

     While roughly treated jujube wood rods are purchased, straight, non-knobs (knotty, gnarl), non-cracks ones are needed. First of all, craftsman goes on scraping beam with plane, till it meet requirement of straightness and shape (fig.1). This procedure needs about 10 minutes. Then, abrade surface of the beam with sand paper to make it smoother (fig.2).


                              fig.1 planing beam/图1 刮 杆


                             fig.2 abrading beam/图2 打磨秤杆

(Two) Making Sleeve

1. Draw a circle on surface around the thicker end of beam and a straight line, which crosses the circle, with a piece of copper on the end of the beam (fig.3).


                       fig.3 drawing a circle and a straight line/图3 划印记

2. Wind a small galvanized iron sheet against an iron beam (fig.4), and shape the sheet into a sleeve (protecting sleeve) by beating two ends of it with an iron bar.


                  fig.4 shaping the sheet into a sleeve/图4 卷帽套

3. Whittle both ends of the wood beam with a plane iron according to the marks made, shaping them into conical contour, then match them with the sleeve by filing the conical contour (fig.5).


                       fig.5 filing conical contour/图5 锉锥面

4. Hoop the thicker end with the sleeve, cut the sleeve’s edge smooth against edge of the beam with steel saw. Trim outside edge of the sleeve with a pair of scissors, leveling it to the end of the beam. Then file the outside edge of the sleeve smooth, it’s not necessary to fix the sleeve to the beam now.

5. Burn a hole diametrically at the thicker end of the beam using a burnt iron pole with square transverse section to install knife of hanging-pan point (fig. 6). Distance between the hole and the end should take 7 portions to ensure U-holder of the knife for hanging a pan may move flexibly round the axis at the end of the beam, if the length of pyramid at the thicker end is divided into ten equal sections (fig. 7).


                           fig.6 position of hole /图6 方孔的位置


             fig.7 position of hole and moving of U-holder/图7 孔的位置与刀承的活动

6. Whittle the thinner end of the beam to install a sleeve. Plane it down and then make it fine with a file. The procedure is similar to the one for the thicker end.

7. Put the sleeve on to the thinner end, temp it firmly till it clamps the end tightly, then fix it with tiny nails. Cut outside edge neatly with scissors and file it smooth then. Decorating effect of copper sleeve popular before is better than present ones.

8. Punch two holes on the sleeve for the thicker end with steel saw and punching tool, making the two hole overlapped with the one on the beam.

(Three)Setting Fulcrum Knife

     This procedure is also called ‘tiao fen-liang’ (挑分量) or ‘jiao fen-liang’ (校分量),namely to fix location for fulcrum knife and install ‘dao-niu’ (刀纽, lifting cord) by experiment。’Fen-liang’ (分量) means weight. ‘Blade’ of fulcrum is actually a certain kind of pin made of carbon steel no.45. Make two edges at both ends of pin by shaving surface of the pin lengthwise to eliminate friction and guarantee sensitivity of the knife. Consequently, the pin acquires the name of “knife”. Sensitivity of pin (or knife) tends to degrade remarkably if pin has a bent surface. As pressure upon edges is high, the edges are easily worn away, both edges and parts contacting with the edges must be processed through quenching treatment. According to stipulations in “Steelyard Testing Regulation JJG17-86”, hardness of knife and U-holder of knife should be above HRC50.[19] While testing knife, a qualitative knife is supposed to have straight edges, no dent on them. Knife should be replaced if it is worn and torn seriously.
Details of this procedure are listed below:

1. Install knife in a hole at hanging-pan point on the thicker end (fig. 8). Hanging-pan point (A) is where weighted object is placed (fig.9).


                           fig.8 hanging-pan knife/ 图8 重点刀


                    fig.9 the distance of 21.5 steps/图9 21步半的距

2. Set “back fulcrum” (i.e. set ‘tou-hao’, 头毫). One ‘step’ (‘steps’,步) is agreed to be distance between two feet of a ‘fen-bu’ (compasses). Adjust the distance of one step till length between knife of hanging-pan point and the middle of the sleeve at end of beam may be properly divided into 22 steps (fig. 10). Make a mark at location one step away point A with a sharp point of ‘fen-bu’. The spot of the mark made is location of back fulcrum (B), i.e. fulcrum for scales of steelyard with 15kg weighed capacity (fig. 9). Actually, the distance from point A to the inside edge of the sleeve on the end of the beam is 21 and a half steps. In this way the mark indicating the maximum capacity would locates neither too close to nor too far from the end of the beam.

                   fig.10 measuring steps to fix a fulcrum/图10 分步找支点

    At first set hanging-pan point (A), then fix front fulcrum (C); after that, following a simple ratio AC=3×AB find back fulcrum (B), according to old craftsman Sun. Another craftsman supplemented, usually AC is about 11cm, AB turns to be about 3.5 cm. This ‘indigenous tip’ gained through experience doesn’t contradict step-measuring practice. Even when different wood is used, the indigenous tip remains unchanged, as craftsmen believe that variety of woods wouldn’t be remarkable.

    At the end of the beam, a short extra span is kept between sleeve and mark indicating maximum capacity. The marks would be abrade and will be reset if they are worn away. Weight of the beam would be less, in turn the short extra span would be compressed; if the original length of span were not adequate, there would no enough space for marks of scale.

3. Install ‘dao-jia’ (刀架, U-holder assembly for hanging pan or hook) to hanging-pan focal knife (fig.11). Punch three rings on complete iron pan. fix three rings to the pan to hang it. Close opening of three rings with a pair of pliers. Every ring ties a pan cord. Weight of each ring is fixed, heavier or lighter ones are forbidden. Hang a ring for pan cord to a ring of U-shape holder of knife.


          fig.11 fixing locations of fulcrum and scale marks through experiment/图11 重点刀架

     To prevent mingle of beam and containing pan cord, the state government stipulates length of pan cord should be two third of length of the beam. Length of moving weight cord equals to half of beam’s length.

4. Set the front fulcrum (‘er-hao’, ‘shou-fen-du’) through ‘tiao-fen-liang’, a so-called ‘clumsy way’. First put on moving weight of 750g, then add weight of 3kg, lift the beam up using a knife with the blade of it upwards as a fulcrum (fig. 12). When the beam achieves almost exact balance (the thinner end of the beam is slightly higher than the thicker one, the point where the blade is is location of front fulcrum (C), i.e. the ‘er-hao’ which is needed. Make a mark here with the knife, take off hanging-pan knife as well as its U-holder.


              fig.12 U-holder assembly for hanging-pan or hook/图12 挑分量

    ‘Tiao-fen-liang’ is the most traditional practice to fix locations of essential points on a beam, i.e. pick out fulcrum or spots of scale marks through experiment. Marks indicating two extreme of scales would overlap on a beam of steelyard, making it look neatly, if this method is taken.

     It’s not easy for craftsmen to take up weight of 15kg with a blade. However, they agree combination of ‘tiao-fen-liang’ and step-measuring is a handy way to fix fulcrum. This method does bypass the difficulty brought up by dead weight of a beam. But, this method may not guarantee exact overlapping of the two marks indicating the extreme. The old experienced craftsman confirms fulcrum may be picked out fully relying on measuring and dividing steps, too.

5. Draw two lines lengthwise on upper and lower sides of the beam.

6. Drill holes on the two fulcrums (B, C) to install U-holder of lifting cord. Drill a vertical countersink and a horizontal hole at C with a bench drill in order to inlay concealed knife (暗刀, ‘an-dao’) and sling(吊毫, ‘diao-hao’) into the beam (fig. 13). Drill a vertical hole at B.


                           fig.13 beam of steelyard/图13 秤 杆

     Well, bench drill and steel saw are not traditional Chinese tools. They are more handy than traditional Chinese drill (舞钻, ‘wu-zuan’), saw and file.

    7. Install outside knife and U-holder at back fulcrum (B). This outside knife corresponds to maximum weighing capacity of steelyard. Shape the vertical hole into a slot with chisel of small size to inlay rest of knife (刀桩). Saw and file a horizontal gutter vertical to the beam to contain knife. Take a piece of iron sheet, bend it curved, cover the vertical hole with it, and nail it to the beam. At the curved cover make a corresponding hole to the hole on the beam by sawing and punching. (fig.14). Put the rest (fig.16, fig.17) taking the knife (fig. 15) into the vertical hole, slip a iron washer onto lower end of the rest that is outside of the lower surface of the beam, then cut extra part of the rest, fix the rest firmly on the beam with a hammer (fig.18). The procedure is completed.


                               fig.14 curved cover/图14 盖 片


                             fig.15 fulcrum knife/图15 支点刀


                          fig.16 rest assembly of knife/图16 刀桩总成


                                  fig.17 rest of knife/图17 刀 桩


                        fig.18 U-holder assembly for fulcrum/图18 支点刀架

8. Install concealed knife and its U-holder at front fulcrum (C). This inside knife is called null point lifting (‘ling-dian-niu’, 零点纽). Even two curved surfaces at both ends of long countersink slot with chisel of small size, befit the inner space to size of sling (‘diao-hao’) (fig.19). Put sling into vertical hole, insert concealed knife into horizontal hole and hole of sling (fig.20). Make a iron cover by cutting and folding nail it on lower part of the beam (fig.21), joint a ring to another hole of sling, closing its opening with a pair of pliers.


                                   fig.19 sling/图19 吊毫


                                fig.20 concealed knife / 图20 暗刀



                                fig.21 curved cover / 图21 盖 片


9. Try installing containing pan and hook at hanging-pan point (A) i.e. joint U-holder assembly with hanging-pan knife. Pan is suitable for bulk goods, hook for hanging.

10. Tie two cord to joining rings of two lifting, making two lifting cord rings. Size of ring is based on size of a palm.

(Four) Setting Scale

    This task is to arrange scale marks on beam of a steelyard, i.e. set graduation of stars (星点分度值, ‘xing-dian fen-du zhi’), including null point. Since woods are not of consistent quality, identical size and equal weight as metals are, each steelyard need to be scaled separately following accounted procedures.

1. Sling the lifting cord of back fulcrum (B) onto a framework, put weight of 15kg onto hook or pan of a steelyard, hang a moving weight of 750g on beam of the steelyard, move it. As the beam achieves balance, the point where the cord tying the moving weight indicates is position of scale for 15kg. Make marks on both sides of the cord by carving two lines vertical to axis of the beam.

2. The lifting cord of back fulcrum (B) remains on the framework, take away bigger weight, keep a weight of 3kg on the pan. Pick out location for the mark of 3kg following the above method, make a sign.

3. The lifting cord of back fulcrum (B) is kept on the framework, put a weight of 5kg onto the pan, pick out the point for 5kg in the same way.

4. Take up lifting cord of front fulcrum (C), empty the pan, move the moving weight. As the beam achieves balance, the position where the cord of moving weight is turns to be the null point, make a sign here. Put on weights of 3kg and 1kg in turn, pick out scale marks for them, make marks.

5. Take the thicker end of the beam with a hand, put the thinner end on
a table, making the hole of knives of fulcrums parallel to surface of the table. Push a pencil to draw a straight line lengthwise on the beam with another hand. Turn the beam 180°, draw another line on the opposite surface of the beam. All marks of scale are distributed on these two lines. The two lifting cords are supposed to locate on these two lines too. Experienced craftsmen may draw a straight line in one push without ruler.

6. Divide these two lines evenly to make scale marks with ‘fen-bu’ (compasses). After trials and experiments, settle a proper span for ‘fen-bu’ (compasses) and make marks with its sharp point (fig. 22). The job is likely to cause presbyopia, if work on it for long time, most craftsmen wear glasses.


             fig.22 measuring steps and making marks/图22 分步、划分刻度

    At line of scale from null point to 3kg., every interval between two marks indicating every 1kg. Should be divided into 10 steps evenly, but craftswoman splits it into 5 steps, every step represents 4 ‘liang’. Then take another pair of compasses (‘fen-bu’), divide the interval into 4 frictions, each unit represents one ‘liang’. In the same way, divide the line on which scale marks of 3-15kg are located. Divide it into 12 steps, then split each interval into 5 sections, each section represents 4 ‘liang’.

     Further setting of smaller unit completes while drilling pores.

(Five) Making Scale Marks

1. Drill pores for scale marks of 0-3kg. With a traditional Chinese drill (so-called ‘tu-zuan’ or ‘wu-zuan’) according to signs made on the beam and the pattern designed in advance for the marks. Make pores on signs carved and make another pore at intuition in the middle of each section made up by every two pores drilled (fig.23). In this way the minimum scale interval represents half ‘liang’. On signs made for unit ‘jin’ drill 6-7 pores on a column vertical to the beam. For a nice look of the beam and easy distinction, this practice is taken.


                           fig.23 drilling pores / 图23 钻孔

2. Drill pores for 3-15 scale marks with the drill according to signs made. Make pores on signs carved and make another pore at intuition in the middle of each section made up by every two pores drilled. Then, the minimum scale interval mark represents 2 ‘liang’. On marks for ‘jin’ make 3-4 pores transversely distributed. On points indicating kilo three more pores, which locate in image of character “pin”(品), should be added on both end of the transverse line of pores.

3. Scale marks, so-called stars of steelyard, made with copper wire. Diameter of copper wire used should match diameter of the pores made. Insert a piece of copper wire into a pore, then cut extra part off against the surface of the beam with a knife (fig.24), after that tamp outside end of the copper wire with back of the knife. All marks (or stars) of scale are made this way. The knife used to cut copper wire should be rubbed against wax, in order to protect the surface of the beam of steelyard. While making steelyard, knife used should frequently be coated with wax (fig.25).


                        fig.24 cutting copper wire / 图24 切铜丝


                        fig.25 rubbing knife against wax/图25 蹭蜡

4. Polish surface of beam of steelyard. polish copper scale marks with wet oilstone to get rid of capillary thorn and make the surface of the beam smooth, neat, preventing pricking (fig.26). Meanwhile woody dust comes up after grinding would stuff crevice around cooper wire, making the copper star fit more firmly. After polishing should check if any copper mark slip off. Dray the wet surface of the beam.


                           fig.26 rubbing stars / 图26 磨 星

5. Nail a trademark plate onto beam of steelyard. It should be placed near the front fulcrum . Information, such as origin, should be noted on the plate (fig.27).


                          fig.27 trademark plate fig/图27 标 牌

6. Waxing. Wax the beam of steelyard several times with a chunk of wax in a hand. Then grind the beam with a string of small links (fig.28). As rubbing produces heat, the wax gradually melts on the beam and seep onto surface of the beam. After this treatment, the surface of it becomes sleek and nice-looking.


                       fig.28 grinding the beam/图28 擦杆面

    Jujube wood is colored , so it’s not necessary to paint it. If beam of steelyard is a white one, paint must be applied to it before waxing. After air-drying, paint a layer of transparent coating to prevent being damped and color loosing.

     Star-patterned scale may be dated back at least to Qing Dynasty. At that time, rulers with scale made by star-setting were popular. Scale of rulers of Ming Dynasty was carved on wood, ivory or bone.

(Six) Assembling

    Assemble pan, hang a moving weight. The moving weight is cast with gray cast iron at foundry shop. The one used today is standardized.

     After procedures accounted above, the steelyard-maker check the production over through himself or herself. If nothing is wrong, a steelyard is sent to another craftsman for testing. Names for parts of an assembled steelyard are illustrated on figure 29.


                         fig.29 Steelyard Drawing / 图29 杆秤总装图
1. moving weight, 2. ring for moving weight, 3. cord of moving weight, 4. (large) sleeve, 5. beam of steelyard, 6. trademark plate, 7. curved cover, 8. rest of knife, 9.U-holder of fulcrum, 10. joint, 11. ring, 12. lifting cord, 13. fulcrum knife, 14. hanging-pan knife, 15. U-holder, 16. joint, 17. link, 18. link, 19. steelyard hook, 20. curved cover, 21. sling, 22. ring, 23. lifting cord, 24. (small) sleeve, 25. ring, 26. pan cord, 27. ring, 28. pan of steelyard.
1.秤砣,2、砣环,3、砣绳,4、(大)帽套(大头箍、保护帽),5。秤杆,6、标牌,7、盖片,8、刀桩,9、支点刀架,10、活洛环,11、连接环,12、绳纽,13、支点刀,14、重点刀,15、刀承,16、活洛环,17、连接环,18、连接环,19、秤钩,20、盖片,21、吊毫(穿在暗刀上),22、连接环,23、绳纽,24、(小)帽套(小头箍、保护帽),25、盘绳环,26、盘绳,27、盘系环,28、秤盘

(Seven) Testing

    A set of regulation for inspecting apparatus for measurement, in which standard of testing and punishing practice were stipulated, had been established as early as the time of the first emperor of Qin Dynasty.[20] In 80s of this century the state puts a regulation of “three fixes” into effect, i.e. fixed weight and length of beam; fixed weight of moving weight, and fixed weight of pan. Fix position of stars and null point according to moving weight. For example, length of a steelyard with weighed capacity of 15kg.is 70cm, and moving weight weighs 750g. It’s convenient to replace a moving weight and other parts of steelyard if they are damaged or missing, after the “three fixed” regulation has been put into practice. When parts of a steelyard are replaced, the steelyard needs to be tested. Without regulation concerning weight of moving weight, while making a new moving weight to replace an old one of a steelyard, casting would be troublesome, as if the new one is heavier, craftsman must wear allowance; if lighter, it would be of no use. Before the “three fixed” was put into practice, there was great variety of moving weight all over the country. It’s easy to cast moving weight of unfixed weight. During period of Republic of China there were two people in office of authority at Tongxian in charge of inspection of steelyard using weights, they were administrated by the local authority, said the old craftsman.

     Weighing Apparatus-Overhauling Station of Tongzhou District is making steelyards according a set of blueprints. A picture of star patterns frequently used is hung on a wall. “Regulation for procedures of steelyard-inspecting” is on a wall of the workshop, the regulation bases on state standard “Steelyard Testing Regulation JJG17-86”. A craftsman is not supposed to test his own products, but steelyard-maker and inspector may work at the same workshop. Following the regulation, inspector should check precision, sensibility and stability, convertibility of a steelyard.

Craftsman Sun’s job includes:

1. Check if star line meets requirement by looking at them.

2. Hang up lifting cord of front fulcrum, put on moving weight, check scale marks for 0-3kg. At first calibrate null point. Then put on weights of 1kg. And 3kg. Separately, see if corresponding scale marks are precise. The Regulation stipulates, error should be lower than 0.8%.

3. Hang up lifting cord of back fulcrum, put on the same moving weight, check scale marks of 3-15kg. Add weights of 3kg, 5kg, and 15kg, see if corresponding scale marks are precise.
Put on weight, beam of a steelyard should achieves a stable poise horizontally. The thicker end (尾,‘wei’, i.e. end) may not be lower than the thinner end (头,‘tou’, i.e. the head) of a beam. Standard for inspection is h, i.e. the vertical distance between the end and the head, may not exceed 1/30 of length of the beam. For example, a steelyard of 15kg, h should be shorter than 2cm. Craftsmen are used to measure the distance using a ruler in stead of the angle made of the beam and a horizontal surface.

     Steelyard and ‘deng-zi’(small steelyard) in use are supposed to have regular check too. Drugstores in Beijing and other area use ‘deng-zi’ to weigh Chinese herb. While precious medicine materials are weighed, balances of small size are used. Every half year a measurement administrative bureau tests their apparatus once, according to staff at a drugstore. They use two standardized weights to check. At present, there is a shop selling and repairing steelyard at Mei-shui-jie Street at Qian-men area in Beijing.

     Steelyards produced in Beijing are sorted into 10 kinds in terms of weighing capacity, including 3kg, 5kg, 10kg, 15kg, 30kg, 50kg, 80kg,100kg, 150kg, and 200kg. Usually, steelyards of various capacity fall into three categories: square pan, circle pan, and hook. A steelyard may either have a pan and a hook at same time, or have only a hook. Hook is used for fish. Steelyard of small size has no hook. If maximum weighing capacity of a steelyard exceeds 30kg, it has no pan, only a hook. A pan is made of galvanized sheet, its shape may be forged according to application. For example, the ones for bulk goods tend to be higher and narrower.

     Units of scale has been changing over long history, but it hasn’t effected the technology and procedure of steelyard-making. Except star pattern, all other things are just same as they were before. In 1958 steelyard with unit of ‘shi-jin’, i.e. ‘jin’ (1 ‘shi-jin’ equals to 10 ‘liang’) appeared. In 1984, the state government started to promote international unit system, requiring making steelyard with unit kilo.

Three. Steelyard-making in Changsha

    Steelyard-making procedures taken by a young craftsman, staff at Hong-yuan Measuring Apparatus Shop in Changshan, Hunnan Province, are recorded below. He made a steelyard, which weighing capacity is 10kg. Mr. Wen Zhifei, master of the young craftsman, gave specific explanation for the technique. The apprentice and old craftsman are relatives.

(One) Planing Beam

1. Plane 1.8 ‘chi’ cubic wood bar into a beam with a plane. Taking the bar in one hand, craftsman takes plane with another hand to scrape (fig.30). Shape the material into a beam with a almost circular transverse section, one end is thicker while another is thinner. At this phase plane the bar roughly first, then scrape it further, when necessary adjust angle of plane iron.


2. File surface with a file, making beam finer.

3. Rub the beam smooth with back of a saw blade.

4. Polish surface of the beam with oilstone, after making the beam wet. Then polish it with water sand paper no. 180.

    Oilstone and water sandpaper are all of modern industry.

(Two) Setting Fulcrum

1. At a point 9cm away from end at thicker part on beam, make a mark (circle a) around the beam on the surface. Cut a groove along the mark on the surface in the section of 9cm. At a point 7.5cm away from the end at thinner end, make same mark (circle b). The young craftsman said, both the locations of 9cm and 7.5cm are rough data picked up randomly.


                           fig.30 planing beam/图30 刮 杆

2. ‘Ka-xian’ (draw a line). Draw a line (line c) in the due middle of the beam lengthwise with carpenter’s ink marker to make ‘zong-bian’ (总边,a line) for scale (fig. 31). This line should pass an end of long axis of a transverse section of the beam.


                        fig.31 drawing lines fig/图31 卡线(画墨线)

3. Starting from circle a to circle b along the ink line take 21 steps and then another 7.2 steps, make marks at two points one step away from circle a, (B, C, fig. 32) The span of compasses is fixed after several trials. The young craftsman said: 21 steps represent 10kg, that is to say, arrange scale marks for 20 ‘jin’ at an interval of 21 and a half steps.


                          fig.32 measuring steps/图32 量 步

4. In the section of 9cm, at the point 1.2cm away from circle a, divide perimeter of a transverse section into 5 steps equally, make five marks (fig.33). The starting point of the circle is located on the ink line. Along the nearest two marks to the ink line, drill a horizontal hole with a electric drill.


   fig.33 dividing perimeter of a transverse section into 5 steps/图33 在截面上分步

5. On both side of point B that is one step away from the starting point A, drill a hole separately (fig.34).


                           fig.34 drilling holes/图34 钻 孔

6. On both side of point C which is one step away from the starting point A), drill a hole separately (fig.35).


                        fig.35 drilling holes fig/图35 钻 孔 图

(Three) Sleeve-making

1. On both sides of the section of 9cm (head) and the section of 7.5cm (end), whittle off wood, file the whittled parts into neat curved surface. Both sections are conical contours, their transverse section is almost a oval. The surface of the section with scale mark line should be left intact.

2. Wax the two sections of 9cm and the 7.5cm with a candle bar, making the surface smooth. Take two cooper sheets, fold the one end of them against a bar of a file, wrap the copper sleeves round head and end of the beam (folded end press the flat one), tie the copper sleeves two rounds tightly with a rope (fixing). Draw the copper sleeves off the beam, cut the two ends neatly. Slip copper piece onto the two ends of the beam, at the spot that is on the extended line of the scale line, punch three pores, fix the copper sleeves on the head and the end of the beam with tiny nails (so-called sesame nails).
Two holes that overlap with the hole on the beam exactly must be made on the copper sleeve on the head part.

3. On part next to sleeve of head of beam cover another piece of copper, fix it with four tiny nails (fig.36)


                 fig.36 fixing a piece of copper with tiny nails/36 钉铜片

4. Put hanging-pan knife into hole A, two ends of the knife are projected outside the beam with the blade upwards. While it’s in use, the knife lies horizontally.

5. Cut edge of the sleeve flat, file sharp edge dull, polish rough part of it. The sleeve would be both good-looking and unlikely to cut hand.

(Four) Installing Fulcrum Knife and Lifting Cord

1. Draw two more ink lines (d, e) symmetrically on both sides of the first one ( c) with ink container, the three lines are almost parallel(fig.37). Scale marks of 0-3kg will be made on either one of the two lines d, e, which is called ‘dan-bian’ (单边,abbreviation is ‘bian’).


                              fig.37 drawing lines/图37 卡 线

2. Saw grooves on fulcrums B and C separately, the grooves cross ink lines d and e (fig.38). Insert fulcrum knife into the grooves. While inserting, smaller end is put forwards and blade downward. The craftsman’s experience is: AB≈2×BC.


                           fig.38 sawing grooves/图38 锯 槽

3. Make two shims with a piece of flat iron wire. These shims are put on knives of B, C separately. Tamp an iron U-fastener down into holes on B, C (fig.39) fold off extra parts of it, hammer the top of the iron U-fastener, knife and shim then would be tamped into the groove (fig.40). Drag out fulcrum knifes, then U-holder and its lifting cord may be put on with them (fig.41).


                 fig.39 tamping U-fastener down into holes/图39 装U形刀桩


                fig.40 hammering the top of the U-fastener/图40 折弯U形刀桩


                       fig.41 putting on U-holder/图41 穿装刀承

4. Tie cords of pan (fig.42).


                       fig.42 tying cords of pan fig/图42 系秤盘绳

(Five) Locating Scale Marks

1. Put the steelyard which has a pan, a hook and a moving weight onto a framework with the lifting cord (立点, ‘li-dian’ or ‘er-hao’) of knife C to pick out null point of ‘dan-bian’ (fig.43). The point where cord of moving weight hangs is the null point, when steelyard poises. Make marks along the cord of moving weight. Weight of the moving weight is 500g.


                  43 picking out null point of ‘dan-bian’/图43 找零点

2. Put 3kg weight onto the pan, move moving weight, pick out a point for 3kg, make mark.

3. Hang lifting cord of fulcrum (支点,‘zhi-dian’ or ‘tou-hao’), 3kg weight remains on the pan, pick out start point (mark of 3kg) of ‘zong’ scale (0~3kg), by same method.

4. Add 5kg weight to hook, sum of weight reaches 8 kg. Take the above procedure find scale mark for 8kg on ‘zong’, draw a sigh.

5. Take off the weight, move away the moving weight, draw out the knife, uninstall the pan and the hook.

6. Measure steps on beam with a pair of compasses, make marks with sharp point of it (fig.44). Adjust span of compasses till it may divide the ink line between marks of 3-8kg into 5 sections symmetrically (5 steps). Then take another two steps and make marks, locations of scale marks for 9 and 10kg are settled.


                           fig.44 measuring steps/图44 分 步

7. On scale line of 3-10kg make marks whose interval represents 1 ‘jin’.

8. Divide scale line of 0-3 kg into 6 steps, each steps represents one ‘jin’. Then divide interval representing one ‘jin’ into 5 sections, each one turns to be 2 ‘liang’.

9. On scale lines of ‘dan-bian’ (0-3kg ) and ‘zong-bian’ (3-10kg), carve marks representing ‘jin’. Marks on point of one ‘jin’ are less remarkable than these on point of a kilo. All marks are vertical to the ink line.

(Six) Making Scale Marks (so-called Scale Stars)

1. Drill pores at marks on ink line for making marks (fig.45, fig.46). First, drill pores on ‘dan-bian’, at intuition and experience drill 4 pores between every interval that represents 2 ‘liang’. The interval between each two pores represents 4 ‘qian’ (20g) then, interval of 2.5 stars is 1 ‘liang’ (50g). At null point, at direction vertical to the ink line arrange three pores. Similarly, at every spot representing 2 ‘liang’, drill 2 pores; spot for ‘jin’, 6-7 pores; every kilo, 8 pores, and at the two end of these lines of pore are three more pores.


                           fig.45 drilling pores fig/图45 钻星孔


                             fig.46 drilling pores/图46 钻星孔

2. Drill pores on ‘zong’. At end drill pores to make image of “3kg”. Starting with 3kg, there are 7 pores at every mark representing a kilo. Depending on intuition and experience, drill 4 pores in every interval representing 1 ‘jin’, dividing each interval of ‘jin’ into 5 sections, every one is 100g. At the spot of 5kg, there are 8 pores as well as three more on each end separately. At the end of ‘zong’, drill pores to make image of ‘10kg’.

3. Filling pores, making scale stars. Melt tin in a pan, add a bit of rosin. At ratio of every 2 ‘liang’ tin for one ‘liang’ mercury, mix them well. Take powder of the combination of tin and mercury, press it into small pores of beam, rub the surface flat, scale stars are finished (fig.47).


                           fig.47 making scale stars/图47 作秤星

Mr. Wen stressed, rosin must be added while mixing tin and mercury. Otherwise, the material won’t stick to the beam. He said, every on ‘liang’ mercury needs 1g rosin, then stars would look nice.

4. Scrape away extra combination of mercury and tin using wet oilstone, meanwhile, polish the beam more smooth. The procedure takes 2 minutes. During this procedure and later on wash the beam in water.

(Seven) Coloring

1. Liming the wet beam to get rid of oil on it (left by hand). After two minutes wash lime powder away.

2. Apply thick black liquid of Chinese herbal medicine ‘wu-wei-zi’ prepared in advance onto the beam with a brush, after 5 minutes wash over the beam, it turns black then (fig.48), air-dry the beam. Heat the beam on open fire or stove to accelerate air-drying. A by-passer probably is familiar with steelyard-making and said, lime is of basic, while ‘wu-wei-zi’ of acid. The young craftsman had no words.


           fig.48 applying liquid of ‘wu-wei-zi’ onto the beam/图48 刷五味子

3. Mop beam dry and clean. Grind sharp edge of sleeve. Error would be brought about if the beam is wet, craftsman Wen said.

(Eight) Assembling and Testing

1. Make two rectangular curved covers (‘gai-pian’) by cutting two cooper scraps, put them separately onto the lower part of the fulcrum points (foot of U-fastener), fix them with four tiny nails onto the beam (fig.49).


                     fig.49 fixing curved cover/图49 钉盖片

2. Assemble the pan, U-holder and their lifting cords. Put on moving weight. Hang the pan onto the hanging-pan U-holder, vice opening of link shut, finish the procedure (fig.50).


                     fig.50 assembling steelyard/图50 安装完的秤

3. Take up lifting cord, put on weights, test steelyard (fig. 51). Null point, 1kg, 3kg and others shown on the beam of the steelyard corresponded to weight of weights. Young craftsman said proudly: Done. OK.


                         fig.51 testing steelyard/图51 试 秤

     Craftsman Wen said, he knew ‘fen-bu’ method taken by his apprentice works too, however, the old way learned from old generation of craftsmen is more reliable. Take a 200kg steelyard-making as example, he accounted the reliable method that handed down from old generation of craftsmen, for locating fulcrum point and scale marks:
       Pick out a location for pan and hook at random, then following below procedures:

First, fix location of fulcrum and range of scale
‘Zong-bian’: put 200kg weight onto pan or hook, locate a spot for mark indicating 200kg ( two fingers away from cooper sleeve), and put on moving weight onto the beam. Suspend the beam of steelyard with a cord, as the beam poises horizontally, the position where the cord is turns to be point for a fulcrum. Draw marks for the first fulcrum and 200kg. Put on weight of 50kg, lift the fulcrum with a cord (‘tou-hao’), as the beam poises, moving weight indicated scale point for 50kg, make marks.
‘Dan-bian’: put on 50kg weight onto pan or hook. Pick out a spot for mark of 50kg(two finger away from cooper sleeve), put moving weight onto beam. Suspend the beam of steelyard with a cord, as the beam poises horizontally, the position where the cord is turns to be point for the second fulcrum. Draw marks for the fulcrum and 50kg. Take away weights, lift the fulcrum (‘er-hao’) with a cord, as the beam of steelyard poises, moving weight indicates point for null, make marks on the beam.
The distances from scale mark for 200kg on ‘zong’ and the one for 50kg on ‘dan-bian’ to the copper sleeve at end of the beam are identical, it’s neat that the two marks are symmetrically located. The young craftsman didn’t follow the tradition of old generation.

     Mark for maximum capacity would be located near to fulcrum and hanging-pan point, beam of steelyard would be shorter, if fulcrum and hanging-pan point are arranged closely next to each other. Actually, a 5kg steelyard’s beam may be either 1.8 ‘chi’ or 2.2 ‘chi’ long. Whatever wanted may be achieved.

Second, setting scale marks of smaller unit.
Based on scale marks made, set scale marks for smaller unit. Pick out scale marks of smaller units following method of measuring steps (‘fen-bu’) with compasses. Adjust interval of a span till one step represents 5kg, divide scale on ‘zong’ and ‘dan-bian’ into 30 and 10 sections separately, make marks with sharp point of compasses. Make clearer sign with carving knife. The minimum unit on ‘zong’ is as precise as 500g, while on ‘dan-bian’ is 2 ‘liang’.

     Beam of this steelyard is 5 ‘chi’ long, moving weight weighs 5kg, maximum scale mark on ‘dan-bian’ represents 50kg, and on ‘zong’ is 200kg, length of sleeve on head and end are 5 ‘cun’ and 6 ‘cun’.
Craftsman Wen said, the old way is best, every time it’s proved reliable. He has been making steelyard for over 40 years. had made no mistake following the old method, no one was defect. If take ‘fen-bu’, the maximum scale mark would miss the good position arranged for it, the beam would look awkward, scale stars even can’t be contained, if scale marks exceed length of a beam.

     Corresponding ratio between weighted capacity, length of beam and moving weight of a steelyard is stipulated in Changsha, Hunan Province. For example, if weighing capacity is 200kg, length of beam and moving weight are supposed to be 5 ‘chi’ and 5kg; if capacity is 10kg, length is 1.8 ‘chi’, moving weight is 500g. Ratio between weighting capacity of a steelyard and moving weight is stipulated as:
3kg 6’liang’
5kg 8’liang’
10kg 1’jin’
15kg 1.5’jin’
50kg 5’jin’
100kg 7’jin’
150kg 9’jin’
200kg 10’jin’

     Besides these series, craftsman Wen makes small steelyards with capacity of 10g and 5g whose beams are of about a hand length. He makes scale following old way, too. Some weighing apparatuses are used to weigh gold. There were also some electric balances and spring balances at his shop, but those are not his products. He believes that steelyard is portable and convenient. Weighting capacity of ‘deng-zi’ used in a drugstore in Changsha is half ‘jin’ (250g).

     Basic structure of a steelyard of Changsha is shown in fig.52. Corresponding terms for main parts of steelyard and used at Tongzhou, Beijing as well as in ancient Chinese books are listed below:
定点(‘ding-dian’): 重点(‘zhong-dian’, hanging-pan point)
支点(‘zhi-dian’, first fulcrum): 头毫(‘tou-hao’)、后支点(‘hou-zhi-dian’);
立点(‘li-dian’, second fulcum):
二毫(‘er-hao’)、前支点(‘qian-zhi-dian’)、首分度(‘shou-fen-du’);
零点(‘ling-dian’, null point):
定盘星(‘ding-pan-xing’, pan-fixing star)、定星(‘ding-xing’, fixing star);
外卡(‘wai-ka’):刀承、刀架(U-holder of knife);
卷篮(‘juan-lan’): 活络环(‘huo-luo-huan’, joint);
秤砣(‘cheng-tuo’, moving weight): 秤锤(‘cheng-chui’)、权(‘quan’).


                     fig.52 structure of a steelyard/图52 秤的结构

Four. Features of Traditional Steelyard-making

    Although there are differences in steelyard-making in Beijing and Changsha, similarities exist too. They all follow some technological rules, set fulcrum and scale through experiment, take method of ‘fen-bu’ and data from practice of ‘fen-bu’, use bought such parts as moving weight, pan, hook and U-holder. Both reflect basic features of steelyard-making in modern China.

(One) Succession of Skill by Demonstration

     Structure of steelyard is simple and tools used to make it are common. Steelyard-making is a craft originated from family workshops. In old days, there were only five steelyard workshop in Tongxian, they are families Liu, Zhou, Wang, Cao and Jin. The station took them over and has performed administration. Steelyards made here are mainly for local consumption. Steelyards made in other areas are also on sales. Vendors in town and local peasants have been using them to weigh agricultural products.

     Usually, a person may make one or five steelyards per day. The station is the largest workshop for steelyard-making in the area, at the prime time there were four craftsmen. Output of production might reach as many as one or two hundred steelyards per month, when demand was large. There was attempt to put the production in assembly line, but the result was not as satisfactory as individual operation. At present, one craftsman and a apprentice may make 6 steelyards each day. There are only about three steelyard-making workshops of small scale left behind, most of them consist of two craftsmen. By 1998, family Liu was the only family workshop survived. All others were gone. Craftsman Sun learned steelyard-making skill at workshop of family Liu, the master he followed was a member of that family. All family members of family Liu were making steelyards. The craftswoman said, her master was the sixth apprentice of master Liu (i.e. last one), the senior fellow apprentice had family name as Mr. Yang, and the second one was master Sun. Masters and apprentices may not be relatives. History of steelyard-making had lasted for over two thousands years, Mr. Sun said, steelyard had been made during period of the great emperor of Qin Dynasty.

     Usually, an apprentice could finish training for steelyard-making in a year by following a master, according to several craftsmen in Tongzhou. A apprentice did simple work or odd jobs first, such as tying cord to pan and moving weight, grinding star marks after stars have been made by old craftsman, then gradually took more complex tasks. They had to do some basic exercises too, including shaping cubic crude material into a knife of proper straightness by filing. All knives used today are bought and ready for next procedure. Planning beam was also a basic practice, which was ranked as the most difficult task for apprentice. Now day’s cylinder beams are shaped into semi-finished products on a lathe, but in old days they were cubic while being purchased. Apprentice practiced on abandoned beams, till he may shape it into a straight beam with one thicker end and one thinner. Started training of new phase after completed one before, the order of training corresponded the one of making, at last apprentice learned assembly. There were no written book or handy proverbs to be relied on, one should only follow words of craftsman and practice, borne them on mind. For example: length of cord tying pan may not exceed 4/5 length of beam, diameter of cord of moving weight may not exceed 1/3 of minimum unit of a steelyard.

     Masters and apprentices were relatives traditionally, they belonged to a same family, according to Master Wen. History of steelyard-making is over 2800 years in China, but the first person who invented steelyard hasn’t been known, he said. He believes, skills of steelyard-making originated from a one person. During old time, craftsmen were ill-educated and did the job following old way. An apprentice should spend two years training traditionally. He learned step-by-step, planning beam first, then fixing copper sleeve. After that, file knife. After two years, a apprentice could make any kind of steelyards, no matter what size it is. At present, the training period is shortened. After mastering shaping a beam, only another 7 or 8 months are needed.
Having taken over old traditional skill from master, apprentice added some knacks found out from practice, formed his own knowledge. They usually knew what to do in order to meet technical requirement, but couldn’t make it clear why doing so. For craftsman of limited education, simple practice is more important than abstract and complex theory. Their intelligence is incarnated on their “golden hands”.
Traditionally, Chinese people attach great importance to morals and ethic. Steelyard-making craftsmen have their own moral standard too. Citing words of master Sun, there is old speaking about scale stars. During the period when 1 ‘jin’ was popularly equal to 16 ‘liang’, stars were signifying seven stars of the Big Dipper (转斗七星) and such characters as happiness, fortune, and longevity(福、禄、寿). Each scale star of that kind of steelyard represented one ‘liang’. Although steelyard is not big, it needs discipline of ethics too. Life span of craftsman would shorten, if he forge defective steelyard to cheat consumer, in other words, if his steelyard short-weighs customers: one year for one ‘liang’. It means, steelyard maker must sacrifice one year of life, if he has no conscience and makes steelyard indicating heavier weight while the real number is less. It’s interesting to put “sky” and steelyard together. Chinese people usually take the Big Dipper (北斗七星) as a sign for distinguishing direction.
In modern measurement industry, steelyard-making is a job of lower class. But craftsman Sun thinks, status of craftsmen is still respectable. Now steelyard is not promoted, the status is less respectable than before.

(Two) Quantitative Rules and Knacks from Experience

    Steelyard corresponds with theory of lever principle, according to modern physics, especially in eyes of an engineer of Measurement Bureau, any kind of steelyard may be made following theory of lever principle. But, craftsmen in Beijing and Changsha haven’t ever applied physical lever principle, instead, they take technical rules out of experience. Although craftsman Sun and craftswoman know that there is a physical principle, they don’t actually use it. Mr. Wen, the master worker in Changsha, even hadn’t ever heard of such a thing. Once mastering effective “practice”, it’s not necessary for craftsmen to learn difficult theory and calculation that seem confusing to them.
The focus for us is how craftsmen in Tongxian and Changsha got number of 21 and a half steps?

     The craftswoman said, she knows formula of lever principle (dead weight is ignored):
W×L = G×m
W stands for weight that is going to be weighed and weight of pan,
L stands for the distance from fulcrum to hanging-pan point,
G stands for weight of moving weight,
m stands for distance from fulcrum to cord of moving weight.

     However, she think it’s too troublesome to calculate with the formula. Here is her explanation of the confusing and complex calculation: ‘Take length of beam, take weight of moving weight, plus weight of pan and weight being weighed, then divide the result got by weight of moving weight.’ She said: ‘it’s really troublesome to get a result after 4 calculation, I tried it myself, but didn’t make it.’ She thinks she’s defect in knowledge, can’t accept the complex method, so found her own shortcut.

     The most traditional practice passed on to apprentice is ‘tiao-fen-liang’ with a knife directly. It means, she fixes location of fulcrums and scale marks through experiment. She mastered this basic method. She had found her own way of measuring steps after long practice, in order to limit tiresome job of lifting heavier weight and to avoid complex calculation. Number of steps is found through following formula (fig.53):
(max. weighing capacity + weight of pan ) / weight of moving weight = steps’ number
Maximum weighing capacity, i.e. maximum of scale, it is 15000g
Weigh of pan, including a pan and a hook, it is 1000g
Weight of moving weight is 750g

     So, (15000+1000)/750 = number of steps (It’s not integer, in stead, is 21and 1/3)


                     fig.53 craftswoman’s calculation/图53 工匠的计算

     The craftswoman said, ‘if only take 21 steps, it’s can’t be divided evenly.’ Namely, length of 21 steps cannot contain 16 kg. As 21×750=15750 < 16000.

     She explained: ‘if take half more, the result would exceed, the result received, 16125, would be more than real capacity (16000). It’s not necessary to carve superfluous section of scale for 125g.’ Her calculation is 21.5×750=16125>16000. Her idea is: Length for 16kg is needed. 15 kg in it is the number of scale. Remainder 125 means the distance between the star of maximum scale mark and the sleeve at end of beam.

     She said: ‘it’s not necessary for me to note weight of beam, neither should I know what quality the beam is of, I do my way then.’ i.e. while fixing fulcrum, dead weight of beam may be ignored; and while fixing major scale marks, craftsmen evade dead weight by experiment. It’s not necessary to recalculate according to lever principle, even if materials of beam is red sandalwood or pear wood.

     Although 21 and a half steps is only a approximate data, they help craftsmen find spots of back fulcrum, bypass complex calculation and troublesome ‘tiao-fen-liang’. This approximate data doesn't decrease precision of scale setting, since locations for scale marks of 15kg, 3kg, 5kg are picked out through experiment.

     The craftswoman assured us firmly that she found her own formula through her practice herself, it was not derived from lever principle. Her master’s calculation was not same as hers. Apprentice takes over master’s practice, and then explores for their own. ‘To craftsmen, the simplest the best.’

     Actually her formula is similar to lever principle. Number of step equals to ratio of distance: (m+L)/L. A section received from the formula, (m+L)/L, is exactly the distance from back fulcrum to hanging-pan point.

     Master Wen in Changsha said, 21 and a half steps and 7.2 steps are not data passed down. Considering practice of apprentice, he said capacity of 10kg, presume 20 ‘jin’, measure 30 steps, 3 multiplied by 7 is 21 (‘jin’), weight of moving weight is 1 ‘jin’ among them, but he can not make the meaning of ‘7’ clear. Therefore, he refereed back to ‘old way’.

     His explanation is more obscure. In fact, young craftsman takes 21 steps and one half steps in stead of 30 steps. Probably he find data, such as 21 and a half and 7.2 on basic of old way.

     Craftsman Wen firmly assured, he knows only old way learned from master, hadn’t learned things like lever principle or scientific knowledge. He emphasizes “practice”, stressing fixing locations on beam of steelyard (such as scale mark for maximum weighting capacity, fulcrum, location of moving weight, as well as the change of the point), didn’t pay special attention to distance on beam of steelyard ( distance between fulcrum and moving weight, fulcrum and hanging-pan point) nor relationship between weight of weighed object and weight of moving weight. He understood, if a weighted object is heavy, the moving weight would drop off beam. What he concentrated on was not distance a moving weight moves over, but the point onto which it moves. He believes scale made in old way is precise. One would make mistake without understanding, if operating according to present ‘theory’.

     Actually, no matter how man calculates, he will finally fix a series of points by measurement. In eyes of craftsman, the most reliable operation is ‘tiao-fen-liang’ and measuring steps. An experienced craftsman may spend less time to measure accurate steps. However, calculation needs mathematical and mechanical knowledge.

     Either in Changsha, or in Beijing, 21 and half and 7.2 steps are only approximate numbers for rough scale-setting. When making scale marks, craftsmen didn’t use the data more. All important scales as basic indicators are all tried out. At present we are not able to be sure how long ago measuring steps may be dated back in history of China.
Having learned craftsmen’s practice, let us have a look at how mathematician in ancient China solved problems concerning steelyard calculation.

Five. Traditional Analysis and Calculation Concerning Steelyard

    There were quite some descriptions about balance and steelyard in ancient books in China, steps articles mainly concerning mechanics are few. Analysis in “Mohist” concerned ‘heng’ and balance is quite typical:[21]
(衡),加重于其一旁,必捶,权重相若也。相衡,则本短标长;两加焉,重相若,则标必下,标得权也。The translation of it is as follows:
The beam…. Explained by: gaining.

     The side of it on which you lay a weight will necessarily decline, because the two sides are equal in weight and positional advantage. If you level them up, the tip will be longer than the butt; and when you lay equal weights on both sides the tip will necessarily fall, because the tip has gained in positional advantage.

     People of later time had different understandings and explanations for this exposition, it’s controversial that if ‘heng’ is unequal-armed steelyard. For example, Qian Baocong believed that the sentences are talking about steelyard, and a character ‘不’(‘bu’), which means ‘not’, should be put in front of ‘相衡’ (‘xiang-heng’).[22] In Graham’s opinion, the first sentence is referring to equal-armed balance, the second indicates moving of fulcrum brings ‘the tip will be longer than the butt’[23]. Dai Nian-zu assumes the sentence is about steelyard, it covers all content of Archimedes’ principle’[24] . In early history scientific knowledge appeared, obscure meaning of terms and different annotation created difficulty to understand early records for people of later time. In any case, there is no substantial material to prove that steelyard had appeared before Han Dynasty, the account of Mohist didn’t give qualitative lever principle.

     Steelyard is a weight-measuring apparatus with precise scale. The unequal-armed beam, unbalance between weighed object and moving weight tend to lead mathematician to make qualitative description about it and calculation.

     During the Spring and Autumn Period, Warring States Period, mathematics gradually turned to be a subject, but relevant historic materials left are less.[25] We are not able to be sure whether multiplication and division in ‘Sun-zi Suan-jing’ (a book on arithmetic, completed aroud 400B.C.) and ‘Xia-hou-yang Suan-jing’(a book on arithmetic, completed before 485B.C.) were applied in calculation of weight-measuring apparatus. Mathematics headed for applied arithmetic during the West Han Dynasty. Problem of fraction and proportion were discussed in ‘Jiu-zhang Suan-shu’ (i.e. Nine Chapters of Arithmetic, completed in 50-100A.C.), but no problems concerning steelyard. Most of traditional mathematical works in China of later time were compilation of solution for problems in application, succeeding layout of Nine Chapters of Arithmetic. [26]

     Zu Geng (was active around 6th century), a mathematician in Northern and Southern Dynasties, compiled ‘Quan-heng Jing’ (Book on the Balance) and ‘Cheng-wu Zhong-lu Shu’ (Methods of Weighing Objects), in that calculation of balance and steelyard was probably be mentioned. But these two books were missing, detail of content are not known today.
Based on experiment, effectiveness of steelyard may be tested easily in practice. This may be used to explain why mathematicians in ancient China did not prove the principle of steelyard mathematically. We are not sure whether the reason why mathematicians didn’t attributed steelyard application into mathematics is its simple application in daily life.

     In mathematical books formed at the end of Ming Dynasty, we finally find some exercises about balance of steelyard.

     Cheng Da-wei (1533-1606) from Xiuning, Anhui province, learnt mathematics from childhood. During twenties he was doing business around lower reach of Yanzi River where handicraft industry and trading were prosperous, meanwhile he visited all erudite scholars. ‘I sorted out meaning of their works, studied their tried algorithms. I have been contemplating over them for over 20 years in my hometown. Once I got inspired, I synthesized various algorithms on my contemplation and added my opinion.’[27] In the 20th year of reign of emperor Wanli (1592), Cheng completed a 17-volume book---‘(Zhi-zhi) Suan-fa Tong-zong’ (General Collection of Algorithms), six years later, he finished ‘Suan-fa Zuan-yao’ (Collection of Major Algorithms).[28]

     During reign of emperor Jiajing, Longqing and Wanli in Ming Dynasty (1522-1619), both businessmen and erudite scholars were respectable in area of counties, such as Shexian and Xiuning. Quite lot calculations in business were collected into ‘Suan-fa Tong-zong’. There were two exercises concerning steelyards on page 48, volume 4 of this book.[29]

Exercise 1

    Now there is a pig. Because there is not a big steelyard, a steelyard of small size has to be used to weigh the pig, but the weight of pig exceeds weighing capacity of small steelyard. The weight of the original moving weight (of small steelyard) weighs one ‘jin’ and ten ‘liang’. When weighing the pig, besides the original moving weight, put on another moving weight that weighs one ‘jin’ and four ‘liang’ and eight ‘qian’, then result indicated on beam of the small steelyard is 67 ‘jin’. How heavy if the pig actually?

Answer: The pig weighs 120 ‘jin’ and 9 ‘liang’ and 6 ‘qian’.

Algorithm: weight of original moving weight is 26 ‘liang’, weight of the second moving weight put on later is 20 ‘liang’ and 8 ‘qian’, sum of them is 46 ‘liang’ and 8 ‘qian’. The sum is multiplied by the number received, 67 ‘jin’, result is 3135.6 ‘jin’. This number then is divided by weight of the original moving weight, 26 ‘liang’, it turns to be 120.6 ‘jin’. 0.6 ‘jin’ may be conversed into 9 ‘liang’ and 6 ‘qian’.

Exercise 2
Weight of an object weighted on a steelyard is 8 ‘jin’ and 2 ‘liang’. As original moving weight of the steelyard is lost, now intend to buy a new moving weight to fit the steelyard, but do not know how heavy the new moving weight. Well, weigh the object mentioned above with a moving weight of 2 ‘jin’ and 5 ‘liang’. Then outcome received is 6 ‘jin’. What is the weight of the original moving weight?

Answer: the original moving weight weighs 1 ‘jin’ and 11 ‘liang’ and 3 ‘qian’.

Algorithm: when the new moving weight is used, the object weighs 6 ‘jin’. 6 ‘jin’ may be conversed into 96 ‘liang’, multiply it by 37 ‘liang’ conversed from 2 ‘jin’ 5 ‘liang’, the product received is then divided by 130 ‘liang’, conversed from 8 ‘jin’ and 2 ‘liang’, result is 27 ‘liang’ and 3 ‘qian’. This is the answer.

     In these two exercises, 1 ‘jin’ is equal to 16 ‘liang’.

     Cheng Dawei collected a lot of mathematical books, ‘synthesized various algorithms, and added my own opinion’. It may be implied from the above sentence that mathematical exercises regarding steelyard may originate from practices of mass of common people and mathematical books circulated among them, he probably add his own research to the collection too. At almost the same time, the European missionaries came to China, and had just found their spot in Guangdong Province. In 1595, Italian Matteo Ricci (1522-1610) went to Nanchang, Jiangxi Province. In 1598, he left for Nanjing. Cheng Dawei’s collection was printed in 1592 at Tunxi, Anhui province, which makes the fact that Cheng was not influenced by European mathematics, obvious. The algorithms described by Cheng derived from traditional Chinese mathematics.

     At beginning of 1990s, Wang Xieshan explained mathematical exercises raised by Cheng with mechanics.[30]

Exercise 1

     Suppose W is the weight of the original moving weight, w is the weight of the new moving weight, G is the weight of the pig, g is the weight gained after weighing, Cheng’s formula goes:
G = g×(W + w)/W

     If suppose a is the distance between hanging-pan point and a fulcrum on the beam, k is the length of interval representing one ‘jin’, two formulas may be created according to lever principle:
G×a = g×k×(W + w)
G×a = G×k×W

     Take off a and k at both sides, Cheng’s formula is received.

Exercise 2

     Suppose W is the weight of the original moving weight, G is the weight of weighted object, w is the weight of new moving weight, g is the weight gained after weighing through the new moving weight, then Cheng’s formula is received:
W = w×g/G

     If suppose a is the distance between hanging-pan point and a fulcrum on the beam, k is the length of interval representing one ‘jin’, formulas may be created according to lever principle:
G×a = W×W×k
G×a = w×g×k

     Take off a and k at both sides, Cheng’s formula is received.

     It’s obvious that the prerequisite for the calculations Wang made is that the “null point” must be located at the exact spot of lifting cord (fulcrum), though he didn’t make it clear in his article.

     Cheng Dawei didn’t explain how he deduced his formula. We can’t be sure either he derived the answer to the exercise from ready formulas, or he summed it up from practices of steelyard-making and application. He didn’t mention relations among dead weight of steelyard, center of gravity, null point and fulcrum. Although weights indicated by scale on beam of the steelyard (numbers of scale marks) were presented, the concept ‘arm of force’, i.e. distances from hanging-pan point to a fulcrum and from spot where moving weight is hung to fulcrum, was not clearly declared. Therefore the numerical value of scale in his calculation was not of an exact concept of distance, but they were equal to those representing distance. Thereby what he focused on was algorithms, not mechanical meaning of the problems. We should note, making procedures may not be the same as mentioned above, if null point of steelyard overlap with fulcrum of it. Craftsman should determine a fulcrum (null point) first.

     If the null point of scale does not overlap with fulcrum, but on a point which is b away from fulcrum, formulas established according to lever principle for exercises would be:
G×a = g×k×(b + W + w);
G×a = G×k×(b + W)
and G×a = W×(b + G) ×k
G×a = w×(b + g) ×k

     It’s obvious that these two groups of formulas may not be deduced from Cheng’s two formulas. Namely, in this case Cheng’s formula would be wrong, at least, they are not precise. When b is a small numerical value, they are approximate formulas.

     The question is, whether we can make it sure that Cheng’s and other old generation’s formulas derived from some equation corresponding with to lever principle, whether it’s possible that mathematicians bypassed lever principle and summed up formulas expressing ratio directly from experience of steelyard-making or application of steelyard?

     ‘Suan-fa Tong-zong’, which was printed in Qing Dynasty and now is kept in Beijing Library, consists of 12-volume ‘Suan-fa Tong-zong’, 9-volume ‘Suan-fa Tong-zong Guang-fa’ and 11-volume ‘Suan-fa Tong-zong Shi-yi’, the latter two were written by Fu Guozhu in Qing Dynasty.

     In “Suan-fa Tong-zong Guan-fa”, Fu deduced three more exercises regarding steelyard from Cheng’s collection:[31]
There is a pig. Use new moving weight of 46 ‘liang’ and 8 ‘qian’, and weigh the pig on a steelyard, then the pig weighs 67 ‘jin’. When use the original moving weight, the pig weighs 120 ‘jin’ and 9 ‘liang’ and 6 ‘qian’. How heavy is the original moving weight?
Answer: 26 ‘liang’.

    If weight of the original moving weight is known, 26 ‘liang’, weight of the pig is 120.6 ‘jin’. The pig would weigh 67 ‘jin’ if the moving weight is replaced with a heavy moving weight. How heavy is the counterweight.
Answer: 46 ‘liang’ and 8 ‘qian’.

    If weight of the original moving weight is known, 26 ‘liang’, weight of the pig is 120.6 ‘jin’. Then weigh the pig using a new moving weight of 46 ‘liang’ and 8 ‘qian’, how heavy is the pig according to indication on beam of steelyard?
Answer: 67 ‘jin’.

     In ‘Suan-fa Tong-zong Shi-yi’, Fu Guozhu concluded three exercises into a versatile formula like:[32]
The dividend, 3135 ‘jin’ and 6 ‘liang’, remains same no matter which moving weight is used to weigh the pig. Therefore, the weight of the original weight is received, when the dividend is divided by weight of weighted object; the weight of the object weighed is received when the dividend is divided by weight of the original moving weight (as in the first case). If the dividend is divided by weight of a new moving weight replacing the original one, a new weight value of weighed object is received; if the dividend is divided by the new weight value of weighed object, the weight of the new moving weight is received (as in the second case).

     Suppose G is weight received by the original moving weight at the first time, g is weight received later. W is weight of original moving weight, w is weight of moving weight used later on. Weight of object weighed and distance between hanging-pan point and fulcrum (arm of force) didn’t change, product of the two is 3135 ‘jin’ and 6 ‘liang’ i.e. (46.8×67 ‘jin’ or 26×120.6 ‘jin’), so Fu’s formula may be put this way:
G×W = g×w

     This new formula may be based on lever principle, but prerequisite remains overlapping of locations of fulcrum and null point.
k×G×W = k×g×w (k represents value of one ‘jin’ indicated by scale mark)

     ‘Dividend remains same’ (原与今同实) equals to ‘identical moment of force’, but concepts of ‘arm of force’ and ‘moment of force” were not clarified after all.

     Cheng and Du didn’t mention ‘null point’, could it be the answer that the two locations of null point and fulcrums are overlapped or they overlooked this? If the steelyard has two fulcrums, and weight of pan is stable, how to make two null points of two corresponding lines of scale overlap their fulcrums?

     Chinese mathematicians continued studying algorithms concerning steelyard in Qing Dynasty (after 1644). Mathematical books complied during prime reign of Kang-xi, such as Fang Zhongtong’s ‘Shu-du-yan’ (1661), Du Zhigeng’s ‘Shu-xue Yao’ (1681) and Tu Wenyi’s ‘Jiu-zhang Lu-yao’ (completed during reign of Kang-xi), included similar algorithms.[33]

     The 26th, 27th, 28th in ‘Su-bu’, the third volume of ‘Shu-xue Yao’ in tile of ‘quan-zhong yi-fa’, ‘er-fa’, ‘san-fa’ (the first algorithm, the second algorithm, the third algorithm of weighing) accounted exercises regarding steelyard, issue concerning location of ‘ping-xing’ (null point) was especially mentioned.[34]

     Quan-zhong yi-fa (the first algorithm of weighing):
Suppose weight of original moving weight is 26 ‘liang’, it can not balance an object. Take another object of 46 ‘liang’ and 8 ‘qian’ to weigh the original object, result is 1072 ‘liang’. What is the genuine weight of the original object?

     Algorithm: 1072 ‘liang’ times 46 ‘liang’ and 8 ‘qian’, product is 50169 ‘liang’. 50169 is divide by 26 ‘liang’, Result is 1929 ‘liang’ and 6 ‘qian’, i.e. the genuine weights of the object.

     Explanation: ratio between weight of the new moving weight and the original moving weight is identical with the one between weight weighed using the new moving weight and genuine weight of the object. Weight of the new moving weight is divided by weight of the original moving weight, result is 1.8. Namely, the new moving weight is 8/10 heavier than the original moving weight. 1072 plus 1072×8/10, we get 1929 ‘liang’ and 6 ‘qian’, i.e. weight weighed by using original moving weight. Multiplication is processed before division in calculation, but result is same.

     Suppose W and w are separately weight of the original moving weight and weight of new moving weight put on later on, G and g are genuine weight of object and weight weighed using the new moving weight. It’s obvious that the first sentence of explanation was wrong: w/W≠g/G. The second sentence may be expressed as:
G = g + g×(w/W -1)
i.e. G = g×W/w

     These are formulas of Cheng and Fu.

    ‘Quan-zhong er-fa’ (the second algorithm) is actually a repeat of Fu’s versatile formula:[35]
Suppose the original moving weight of a steelyard if lost. There are two objects, the heavier one is 1929 ‘liang’ and 6 ‘qian’, the lighter one is 46 ‘liang’ and 8 ‘qian’. Weigh the heavier one on a steelyard using the lighter one as moving weight, the result is 1072 ‘liang’. How heavy should original moving weight of the steelyard be?
Algorithm : 46 ‘liang’ and 8 ‘qian’ is multiplied by 1072 ‘liang’, product is 50169 ‘liang’ and 6 ‘qian’. The value is divided by 1929 ‘liang’ and 6 ‘qian’, result is 26 ‘liang’, i.e. the solution.
Explanation: Relation between 1929 ‘liang’ 6 ‘qian’ and 1072 ‘liang’ is same as relation between 46 ‘liang’ 8 ‘qian’ and weight of original moving weight. So answer is received through multiplication and division.

     ‘Quan-zhong san-fa’ (the third algorithm) brings the issue into deeper level:[36]
Suppose the original moving weight of a steelyard is lost. There are two objects whose weights are unknown are weighed on a steelyard. Weight of the heavier one turns to be 52 ‘liang’ if using the lighter one as moving weight; weight of the lighter one turns to be 13 ‘liang’ if using the heavier one as moving weight. How heavy should original moving weight of the steelyard be?
Algorithm: Product found by multiplying two numerical value each other is 676 ‘liang’. Then calculate for square roots, which is 26 ‘liang’. Answer is received.
Explanation: Middle ratio of two numerical values is weight of the original moving weight. Way can be found through calculation for square roots of the product of two values. Another method is weigh an object with another object of same weight as the weighed object, indication shown on steelyard is weight of original moving weight.

     If ‘ping-xing’ (null point) and ‘ti-suo’ (fulcrum) are located on same point of beam of steelyard, results received following the above three methods are approximate numerical values. If they are located on different points, the bigger the distance between the two, the bigger the error of measurement would be. And even received result is quite different from genuine weight. This should be noticed.

    ‘Middle ratio of two numerical is weight of the original moving weight’ (两数之中率, 即原锤之重) is a common formula to find out result, writer didn’t explain how it was received. It’s not difficult to justify its rationale according to lever principle.

     Suppose G and g are weight of the two objects weighed, W is the weight of the original moving weight, a is distance between hanging-pan point and fulcrum (arm of force), b is interval between every two scale marks that are close to each other. When weigh the two objects taking them as moving weight for each other separately, and null point overlap with ‘ti-suo’ (fulcrum), followed equals are received:
G×g×b = g×a
G×g×b = G×a

     Use the original moving weight to weigh weight of 1 ‘liang’, create another formula: 1×a = W×b, put the above three equals together, deduce formula of Du Zhigeng:
W = √G×g, as G = g, W = G

     “Ti-suo” represents fulcrum (or lifting cord). Wang Xieshan believes “ping-xing” is the gravitational center of beam of steelyard; when the beam is supported at the gravitational center without external force, steelyard poises, so the gravitational center is called ‘ping-xing’.[37] In our opinion, “ping-xing” stands for null point, or the hanging-pan point of system made of beam of steelyard and pan, hook as well. Du’s formula works only in case that null point and hanging-pan point of the system are all located on the point at which the ‘ti-shou’ (fulcrum) is. It’s logical that explain ‘if they are located on different points, the bigger the distance between the two, the bigger the error of measurement would be. And even received result is quite different from genuine weight’ with opinion that ‘ping-xing’ is null point. The tiny distance between ‘ping-xing’ and ‘ti-shou’, was supposed made by errors in overlapping of the two location and scale-making.

     Du Zhigeng’s description was more comprehensible than the Cheng’s and Fu’s, reflecting Chinese mathematicians’ understanding about the issue.

Six. European Algorithms for Lever Problems Introduced in the 17 Century

     New exercises were not raised when European algorithms about lever principle was introduced into China. The new way (proportion algorithms) was used to solve the traditional Chinese exercises, i.e. the exercises Cheng had calculated before.

     Eight-volume ‘Tong-wen Suan-zhi Tong-bian’, translated and complied by Metteo Ricci and Li Zhizao (1565-1630), which was printed in 1613. Content of this book mainly derived from Epitome Arithmeticae Praticae (1585) compiled by C. Clavius (1537-1612) and practical problems on ‘Suan-fa Tong-zong’.[38] Followed two exercises were in a section named ‘bian-ce-fa’.[39]

     There is an ivory. There is no big steelyard that is big enough to weigh the ivory, the weight of the ivory exceeded weighed capacity of a steelyard of small size. So when weighing, besides the original moving weight of small steelyard, put on new moving weight, then result indicated on beam of the steelyard is 67 ‘jin’. Data known are weight of original moving weight, 1 ‘jin’ and 10 ‘liang’; the new moving weight, 1 ‘jin’ and 4 ‘liang’ and 8 ‘qian’; then how heavy is the ivory according to the small steelyard.

     Algorithm: Sum of weights of the original moving weight and the new moving weight is first item of a proportion, the second item of the proportion is the result of measuring, the third one is the weight of the original moving weight. The first, 46 ‘liang’ and 8 ‘qian’ ; the second, 67 ‘jin’ ; the third, 26 ‘liang’; the fourth, 120 and 3/5’ jin’.

     Weight of an object weighted on a steelyard is 8 ‘jin’ and 2 ‘liang’. As original moving weight of the steelyard is lost, intend to cast a new moving weight to fit the steelyard, but do not know how heavy the new moving weight. Now weigh the object mentioned above with a new moving weight. Then outcome received is 6 jin. What is the weight of the original moving weight?
[Algorithm:] At first unify unit of all data by conversing them into “liang”. Weight of the object balanced by new moving weight, 96 ‘liang’, is the first item of a proportion; the second item of the proportion is weight of new moving weight, 37 ‘liang’; the third one is weight of the object balanced by the original moving weight, 130 ‘laing’, so the weight of the original moving weight can be received. Converse result into number of unit of ‘jin’, after calculation. The first, 96 ‘liang’; the second, 37 ‘liang’; the third, 130 ‘liang’; the fourth, 27 ‘liang’ and 3 ‘qian’ and 3/13 ‘fen-qian’ (namely, 1 ‘jin’ and 11 ‘liang’ and 3 ‘qian’ and 2 ‘fen’ and 7 ‘hao’, remainder exists).

     Except “pig” was replaced with “ivory”, the problems and data of two exercises were identical with the exercises in ‘Suan-fa Tong-zong’ (See pp.29-30). Nor did Li and Matteo Ricci use concept of ‘distance’ (arm of force). The difference between two books is, Li and Matteo Ricci didn’t follow Cheng’s algorithm to solve problems, in stead, they took European solution based on proportion. Matteo Ricci was supposed to be versed in European method based on proportion, while Li were familiar with method in ‘Suan-fa Tong-zong’. They may sum up the two solutions, and made them into words above.

     The actual work which introduced European lever knowledge from angle of mechanics were ‘Yuan-xi Qi-qi Tu-shuo Lu-zui’ (1627), a collection translated and compiled by Johann Terrenz (1576-1630) and Wang Zheng (1571-1644). The difference of their work from other traditional Chinese technological books is that they noticed theory behind technology, i.e. ‘surveying and mathematics’ (度数之学).[40]

     The second volume of ‘Yuan-xi Qi-qi Tu-shuo Lu-zui’ introduced lever principle and European algorithm with illustration, establishing concepts of力 (force), 重(gravity or weight), 分 (distance): section 9-15 ‘explanation on for balance’, section 16-33 ‘explanation for deng-zi (steelyard)’, section 34-48 ‘explanation for lever’ (fig. 20). ‘Explanation for balance’ was dealt equal-armed balance, concentrating on structure and hanging-pan point of balance. Steelyard was convenient for weighing, while scale was precise, according to the section 25. But the explanation for it seemed awkward. ‘Explanation for lever’ was on common application ( such as crowbar) and calculation of ‘force’ (weight or ability), basically, it was an expansion of ‘explanation for deng-zi’.

    ‘Explanation for deng-zi’ was dealt issue of balance of lever, including calculation of weight (gravitational force) and distance (arm of force). It was told: ‘deng-zi’ (steelyard) had two components, one was beam, another was ‘ti-xi’ (lifting cord) (section 16). ‘Niu’(纽, lifting cord) was supposed to be ‘ti-xi’ ( section 21). ‘Niu-xin’ (纽心) should represent edge of fulcrum knife for steelyard (section 22). ‘Zhong’ (重) stands for both object of weight (so-called ‘zhong-ti’, 重体) and weight (section 23). Dead weight of lever is a calculable factor for balance (section 23). ‘Deng’(等, equal) or ‘zhun-deng’ (准等, actually equal) of ‘liang-zhong’ (两重,two objects) on ‘deng-zi’ means poise of objects of different weight on both sides of lever, which was not identical as ‘xiang-deng’ (相等,equal) of ‘liang-zhong’ (两重, two objects) on balance (section 17). ‘Deng-liang’ (等梁, equal beam) represented a beam in balance.(section 23)

     Lever principle was described in volume 2 in proportion, stressing ‘it’s foundation of science of weight, different algorithms originated from it (fig.54):[41]
     When a beam with two objects whose weights are not same poises, ratio between weight of heavier object and the lighter one equals to ratio between lengths of longer arm of force and the shorter one of beam.
‘Yi’ (乙) weighs 8 ‘jin’, ‘jia’ (甲) weighs 2 ‘jin’, ratio between them is 1:4. When beam with them on each end separately poises, the interval from fulcrum to point ‘wu’ (戊) on the beam consists of 4 sections, the interval from fulcrum to point ‘bing’ (丙) is of one section, the ratio is 4:1 too. The two ratios are identical.
A piece of qualitative explanation was added in the 20th section in volume 2: ‘the heavier and farther from fulcrum is the object on the long part of beam, the faster falls it’. Then the writer listed 13 examples of calculation. The 26th went like (fig.55):[42]
Two objects hung separately on either end of a beam, when balance is achieved, where is the fulcrum located?

     ‘Jia’ (甲) weighs 6 ‘jin’ on point of ‘ding’ (丁), ‘yi’ (乙) weighs 2 ‘jin’ on point ‘wu’ (戊). The beam is divided into 4 equal sections. Where is the fulcrum located?

     Sum of weight ‘jia’ (甲) and ‘yi ‘ (乙) is 8 ‘jin’. Take European method of ratio.
the first, 8’ jin’, sum of weight of ‘jia’ (甲) and ‘yi’ (乙)
the second, 2 ‘jin’, weight of B
the third, 4 sections, whole length of the beam
the fourth, 1 section, distance between points of ‘ding’ (丁) to ‘bing’ (丙)
    So the fulcrum required is located on point ‘bing’ (丙).

     From section 41 of volume 2 there appeared concept of 能力 (ability).     The term stands for effect of application of force relevant to the change of point of application of force or arm of force (section 43) or intensity of force ranked with unit of weight. (section 46).

     While proceeding explanation for pulley, wheel and spiral, the writers applied lever principle, but he didn’t continue this principle to explain the 54 machines in volume 3.

    ‘Yuan-xi Qi-qi Tu-shuo Lu-zui’ was mainly concentrating on issues regarding statics, didn’t touch topics, such as trajectory and falling body, that represented new trend of European mechanical knowledge. This was probably because Wang Zheng preferred to applicable technologies.
Mathematicians in Qing Dynasty proceeded European algorithms when they solved problems relevant to steelyard. During reign of Kang-xi, Fang Zhongtong picked out two exercises from ‘Suan-fa Tong-zong’ and put them into ‘su-bu’, the 22nd volume of ‘Shu-du-yan’, gave tiles as ‘jiao-cheng-shi’ and ‘jiao-chui-shi’.[43] Difference from the original ones was that he took algorithm using ratio as in ‘Tong-wen Suan-zhi’ and ‘Yuan-xi Qi-qi Tu-shuo Lu-zhui’. Mathematicians of later ages still took over the European algorithm to solve traditional Chinese problems relevant to steelyard.


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