Editorial notes and correction to the discursive part

Li Rui added 16 editorial notes in his edition pointing out what corrections were made to the material he was using for his edition. I noticed that for some of the sentences, he signals modifications he made to the text, while the same text remains identical in both editions of the Siku quanshu. If Li Rui says that he corrected some characters in one sentence, I would expect to find the sentence without any correction in the Siku quanshu.

For example, when Li Rui writes that he suppressed the character “to have”, you, 有;

this character is in both editions of the siku quanshu and was suppressed from the edition by Li Rui. There are differences between the editions and these differences are signalled by an editorial note. That is what happened for most of the cases, but concerning three cases, I could not see any differences between all the three editions. For example, in pb 3 and 39, Li Rui writes that two characters are missing in the diagram, and the characters are not

67 [Mei Rongzhao, 梅荣照, 1966]. p. 111.

26 missing in any of the editions. In pb 38, Li Rui said that the charcter wei, 為, is wrong and that he suppressed it. There are no character wei in any of the Siku quanshu I consulted [see Table 1]. The editorial notes written by Li Rui do not always match with the text copied in the different editions of the Siku quanshu.

Another observation is interesting concerning editorial notes. Another editorial note, this time written by the editor of Siku quanshu, is also significant. In the pb 53, the editor of Siku quanshu, mentions that he replaced the square in the diagram by a rectangle to make the reading easier, and the diagram in Siku quanshu edition contains a rectangle (part containing 甲 in Figure 1,). In Li Rui edition, although we can read the commentary of the Siku quanshu, the diagram contains a square (idem in figure 2).

Figure 1

Figure 2

After consulting the different editorial notes, we will now turn to examine other differences which are not signalled in editorial notes. Thoses are presented in Table 1 and 2 in the supplement of this study. I could also notice a total of 95 other differences between the edition by Li Rui and the two Siku quanshu. [See table 2]. In 73 cases, WYG and WJG are the same. In only 3 cases, the three editions show three different versions (pb. 4; 6; 58). For 13 cases, WYG and LR are the same, and in the 6 other cases WJG and LR are the same.

27 The copy in the WYG is more reliable than the WJG. The WJG contains in fact several mistakes which seems be due to dictation: like 圓, “circle”, instead of 元, “original” (pb.43);

or 十, “ten” instead of 實, “dividend”, (pb. 48). Some are also graphical mistakes, like: 十,

“ten”, instead of 千, “thousand” (pb.10), or 去, “to go”, instead of 云, “to say” (pb.23), differing of only one stroke. For my purpose, I take into account only the other 73 cases mentioned above, that is when the WJG and WYG are identical and present differences with the edition by Li Rui.

Among the 73 cases, 35 cases are due to synonymy, for example: 原 instead 元, which both are pronounced yuan and means “original”⁶⁸; 以 instead of 依, both pronounced yi and translated as “according to”. Or they are due to syntax divergences, like 一十八 in Siku quanshu instead of 十八 in LR, which in both cases means “eighteen”. We cannot consider that, in these 35 cases, one of the editions is mistaken. But among the 73 cases, the other 38 cases are indeed due to syntaxical or vocabulary mistakes. In 8 cases the Siku quanshu is correct and LR is wrong. Therefore, there are 30 cases for which LR is correct and the Siku quanshu is wrong, for example, in pb.25, one reads 167 in Siku quanshu instead of 176 in LR (pb.25), or 古徑, “diameter according the ancient lü” in LR instead of 方徑, “side and diameter” in the Siku quanshu (pb.43), the later being incorrect in the two copies of the Siku quanshu. We also notice that a sentence in pb. 40 is in LR and is missing the two Siku quanshu, while the same kind of disparition in pb.34 is signalled by an editorial note. Among these mistakes, 16 of them are corrected by an editorial note by Li Rui as we saw. So the question is why Li Rui did not write editorial notes about the 14 other mistakes?

From these evidences, it seems that Li Rui is doing two things. Li Rui first responds to the commentary of the editor of the siku quanshu concerning the difference between the Celestial Source and “Borrowing the root”, and secondly, he is doing a critical work of the materials (on discourse and diagram) he is using. It is impossible to answer definitvely to the question of the identification of the materials. He could be using a copy of the Wen lan ge which discourse contains differences with its original source in the Siku quanshu. He could have access to the “original” edition of the Wen lan ge, or may be to another edition of the Siku quanshu. It could be also that he consulted the Yongle dadian or a copy of it, whether we cannot say if he had access to it. Or may be Li Rui copied and corrected the oldest edition of the Yigu yanduan he could find and inserted his commentaries and the ones of the siku quanshu inside, or at least he collated at least two editions available to him. But it seems that Li Rui might have in hand two different editions of Yigu yanduan when he prepares the Zhibu zuzhai congshu edition. At this step of the study, speaking of the Siku quanshu as a uniform and unique book vanishes. Now we will see that it becomes difficult to apply the qualificative of “original” to any of the source.

68 Here, we did not compare the commentaries in the Siku quanshu with their copy in Li Rui’s edition. The character yuan in question here does not concern the case presented in a later paragraph. This later paragraph is dealing with the use of this character yuan in commentaries.

28 A second detail is worthy to be mentioned. In his editorial notes, when Li Rui signals corrections, he allways refers to the previous edition of the text written by Li Ye by systematically naming it yuan ben, 元本. The commentary of Siku quanshu uses another character to refer to text which is copied. The character, yuan, 原, is used by the editor of Siku quanshu in pb. 53, 54, 61, 63. Both characters, 元 and 原, can be translated by

“original”. After checking others commentaries of different others mathematical texts that are compiled in Guo Shuchun collectanea, it seems that the character yuan 原, is normally used to refer to text which is copied indinstinctivly of it provenance⁶⁹. Here “original” means the material used as sources. In other texts whose commentaries are attributed to Li Rui⁷⁰, like the 疇人傳⁷¹, Chourenzhuan, the editorial notes use the following characters: yuan ben, 元本, for yuan dynasty editions, ming ben, 明本, for ming dynasty editions, and yuan ben, 原本, for the source editions which copied⁷². Li Rui in his commentary of Ceyuan haijing⁷³ also systematically uses the character yuan, 元, for mentioning the same kind of corrections, and one knows⁷⁴ that Li Rui’s edition of Ceyuan haijing is not only based on the siku quanshu but also on a manuscript referred as Yuanchaoben, 元抄本⁷⁵ which belonged to a certain Ding Jie, 丁杰. This manuscript contains the print of the seal of Ruan Yuan who provided documents to Li Rui⁷⁶. Mei Rongzhao estimates this manuscript to be written between 1310 (Yuan dynasty) and 1381 (Ming dynasty) and this manuscript is actually still available in Beijing National Library. Would it be impossible that the edition of Yigu yanduan followed a similar path and that Li Rui is consulting an edition dated from the Yuan dynasty or at least that he is attributing to the Yuan dynasty?

I am aware that such arguments are not enough to show that Li Rui is not reading the Yongle dadian or a copy of the Wen lan ge, but these arguments are sufficient to question the origin of the sources of the available editions. 49052 characters compose the Yigu yanduan, the discrepancies between editions concern only 95 characters. At least, I can deduce from the meticulousness of Li Rui’s editorial notes that his edition might be quite faithful to an older edition of Yigu yanduan, whatever this previous edition is. Nevertheless we can admire the precision and the quality of the work of the editors of the Qing dynasty.

29 2.2.2 Treatment of diagrams.

This quality of work can be also admired in respect to the edition of diagrams, although this work is not visible at first sight. As the present study relies on a collection of diagrams, I wanted to check how reliable the diagrams provided by the current editions are.

I compared carefully the two editions of the Zhibuzu zhai cong shu and the wenyange Sikuquanshu, and it appears that there are very few differences concerning the shape diagrams. Making exception of the diagram of pb.3, when there are differences, those are explicitly mentioned by commentators, like in the example above. I observed that in fact the two editors kept exactly the same proportion in reproducing diagrams.

The page format of Wenyange edition of the Siku quanshu is bigger than the Zhibuzu zhai cong shu. The first one is a manuscript, while the second one is printed with woodblocks. A woodblock contains the script of a whole page. After measuring the dimensions of each diagram reproduced in the statement of problems in the original edition of the Wenyange siku quanshu preserved in the National Palace Museum in Taiwan, I noticed that those diagrams were constructed in order to be proportional to the data presented in the statement of problems.

Let’s examine a sample [table I] of figures issued from the same category of problems: a square pond inside a circular field. On this sample, one can see that the dimensions of the square are changing from one problem to the other, while the circle keeps the same dimensions.

30 Table I. samples of measurements and data from diagrams in the statement, wenyange edition of Siku quanshu.

problem Diagram presented in Wenyange siku quanshu

Data in the problem

Measurement in millimeters

15 Diameter: 120 bu Diameter: 52 mm

Side: 52 bu Side: 22 mm

16 Diameter: 72 bu Diameter: 52 mm

Side: 18 bu Side: 13 mm

17 Diameter: 54 bu Diameter: 52 mm

Side:24 bu Side: 23 mm

20 Diameter: 60 bu Diameter:52

Side: 15 bu Side:13

A cross product for each case shows the measurements are proportional to the data given in the statement. The same operation was done on every diagram reproduced in the statement in the edition of Wenyange siku quanshu: all diagrams are proportional to data. I compared these measurements with other ones taken in others editions of the Siku

31 quanshu and of the Zhibuzu zhai congshu. The same observation can be made, although there is a loss of accuracy in the Zhibuzu zhai congshu. May be one can attribute the latter to technique of carving used in blockprinting. One notices that while the editions of the siku quanshu and Li Rui edition have different sizes of pages, the shapes of the diagrams are the same and they keep the same proportions: Li Rui’s edition looks like a reduction of the siku quanshu and diverse corrections that were added to diagrams show that the editors paid lot of attention to the dimensions.

If there is work of the editors on the proportion of the diagrams in the statement of problem, the process in less clear concerning diagrams in the Section of Pieces [of Areas] as one will see in a later example (pb.38). The difference between the Wenyange Siku quanshu and Li Rui’s edition concerning the diagram in Section of Pieces [of Areas] of the problem 3 is on this point significant. One notices that the side of the expanded square and the diameter of the expanded circle are exact representations of the dimensions of the diagram in the statement multiplied by 1.4, the latter being proportional to data, in the Siku quanshu [Figure.3]. In both [figure.3] and [figure.4], the expanded circle is represented by dotted lines. But in Li Rui edition, the central circle is smaller. Actually, the dimension of this circle is the dimension of the circle given in the statement reduced exactly by 1.4 [Figure 4]. The consequence is that instead of having squares marked by dotted lines, one has rectangles (part containing the character 從). This mistake is not the result of a problem of carving or technical drawing. It is due to computation. This mistake in the edition by Li Rui shows that there is practice of measurement for publication of mathematical figures.

Figure 3

Figure 4

32 One can see that proportions are respected in diagrams, and that diagrams are corresponding to data of wording and to solutions. Although proportions are respected, ratios are different from one problem to the other and no general law or pattern of repetition among the whole list of diagrams can be observed concerning these ratios. But I notice that in the wenyange among the 65 diagrams placed in the statement, 51 of them, representing a geometrical figure inscribe in another figure, measure 52 mm. The 14 other diagrams have different measurements due to their shape: two geometrical figures next to one another that are connected in one point. The dimension of 52mm is probably an editorial constraint, and the other dimensions are computed and drawn according to this latter.

Concerning the second type of diagrams presented in the section of pieces of area, it is difficult to make the same observation. Many of the diagrams represent area multiplied by big numbers, or results of combination of different areas. But for diagrams which resemble the one given in the statement, dimensions are the same. Here follows the diagrams of problem 38. The diagram placed on the right of the page is given with the following commentary by Li Rui: “the diagram on the right is wrong. To correct the signification (意), I added [a diagram] on the left. [The diagram on the right is wrong]

because the black [line] stands for the original [area]. If one asks [the lengths and the widths of] the dry and the water fields, then a dotted [line] should stand for the original [area].

When one subtracts one piece that is the difference of the two widths, the piece “subtracted by going” and the piece “subtracted by coming⁷⁷” should be equal. They are both the difference of the widths multiplied by the width of the dry [field] at the small [part] at the bottom of the genuine area.”⁷⁸

At first sight, the two diagrams look quite similar.

77 減來, jian lai, “to subtract by coming”, 減去, jian qu, “to subtract by going”.

78右圖舛誤. 以意訂正如左. 葢黑者為元. 問水旱田, 點者元. 減一段, 即二闊差, 昇去減一段與來減一段等.

並是闊差乘旱闊底小真積也.

33 A redrawing will help to see the differences.

Figure 5. Left: “original” diagram⁷⁹

79 j: subtract. j1: subtract the original [area]. j2: subtract by going. j3: subtract by coming. s1-2: water [field]. d:

dry [field]. n: five. The number five is not in the WJG and WYG siku quanshu. n1: this part is difficult to read, either it is “twenty”, or it is “ten” and the stroke which should be above is draw inside the number. n2: ten. a:

length of the water field, seventy five bu, as the divisor. b: difference between the width and the length of the dry field, thirty five bu, as the divisor. c: width of the water field, twenty five bu, as the divisor.

One notices that the line a+b+c is represented by a thick band.

34 Figure 6. Right: diagram with correction by Li Rui.⁸⁰

I will not detail the procedure here, and I will just point out the corrections made by Li Rui. Li Rui is correcting several parts on the new diagram: he is changing a black line into a dotted line. The two areas marked by “jian lai” and “jian qu” are drawn the same size. One of the numbers 5, named n in my caption, at the bottom is placed differently, just under a small stroke. This stroke is significant; it is the symbolic starting point of the width of water field, which is required for the computation of the polynomial. This number n, which is visible in the edition by Li Rui is not in the editions of the Siku quanshu I consulted. And the way the stroke between n and n1 is placed in the original diagram lead to ambiguity in the reading of the caption. It is not clear if it is a geometrical stroke or a part of the character 二, er, “two”. When Li Rui corrected this diagram, he not only added some modifications in the caption and graphic, but he also changed the dimensions. Those reproduce exactly the ones of the diagram given in the statement, which later is proportional to data.

The editor of the Siku quanshu has a completely different attitude concerning this diagram. He mentions that the diagram has a problem and proposes a totally different diagram for a different procedure and says that this procedure is conforming to the “old procedure”. This behavior of the commentator happens several times. It is not the only case when diagrams are reconstituted and added by the editor of the siku quanshu. The problem 56 is one of the problems favored by historians⁸¹ to understand the “old procedure”

because it is provided with a diagram title “舊術又法圖”. But the commentator writes: 今增

80 j: subtract. j1: subtract the original [area]. j2: subtract by going. j3: subtract by coming. s1-2:water [field]. d:

dry [field]. n: five. n1-2: ten. a: length of the water field, seventy five bu, as the divisor. b: difference between the width and the length of the dry field, thirty five bu, as the divisor. c: width of the water field, twenty five bu, as the divisor.

81 [Xu Yibao, 1990], p. 71.[Kong Guoping, 1987], p.110 p. [Kong Guoping, 1999], p. 199

35 一圖義於後而舊術又法 “Now, I added one diagram and a meaning followed by an old procedure”⁸². The title of the diagram quotes characters from the commentary and is clearly an addition by the Qing dynasty commentator. This shows that the commentators were both versed in the procedure of Section of Pieces [of Areas], but have different attitude. Li Rui corrects the diagram, while the editor of the Siku quanshu supplements it without correcting. One will see that these attitudes can also be differentiated in the writing of polynomials, and that this had consequences on the way the text written by Li Ye was interpreted.

This part of the study shows that there is a practice of measurement in editing mathematical diagrams in the Qing dynasty. It also shows that editors were well acquainted with the procedure of Section of Pieces [of Areas]. They were able to differentiate what Li Ye

This part of the study shows that there is a practice of measurement in editing mathematical diagrams in the Qing dynasty. It also shows that editors were well acquainted with the procedure of Section of Pieces [of Areas]. They were able to differentiate what Li Ye