Abstract
The texts of algorithms fall under the general rubric of instructional texts, discussed by J. Virbel in this book. An algorithm has two facets. It has a text—a written text—, which usually appears to be an enumerated list of operations. In addition, whenever an algorithm is applied to a specific set of numerical values, practitioners derive from its text a sequence of actions, or operations, to be carried out. In the execution of the algorithm, these actions generate events that constitute a flow of computations eventually yielding numerical results. This chapter aims mainly to develop some reflections on the relationship between these two facets: the text and the different sequences of actions that practitioners derive from it. I use two tools in my argumentation. Firstly, I use the description of textual enumerations, as developed by Jacques Virbel, to find out how enumerations of operations were carried out in the text of algorithms and how these enumerations were used. Then I focus on the language acts carried out in some of the sentences composing the texts, since, when prescribing operations, the texts of the algorithms differ in that they use distinct ways of carrying out directives. The conclusion highlights different ways in which the text of an algorithm can be general and convey meanings that go beyond simply prescribing operations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In fact, an algorithm can even integrate means for carrying out different tasks. An example from ancient Chinese mathematical sources is described in Chemla 1997a. However, I’ll leave this aspect aside here. The completion of this article was supported by the Chinese Academy of Sciences Visiting Professorship for Senior Foreign Scientists 外国专家特聘研究员, grant number 2009S1–34 (Beijing, 2010).
- 2.
See J. Virbel, “Textual enumerations,” Chap. 6, in this book. This chapter could not have been written without the joint research work carried out over the last few years with the authors of the other chapters in the book and the various colleagues who took part in the seminar “History of science, history of text.” It is my pleasure to acknowledge my intellectual debt to this collective endeavor as well as to Ramon Guardans for his stimulating reactions. I am also glad to express my gratitude to Richard Kennedy for his help in preparing the final version of this text.
- 3.
See J. Virbel, “Speech Act Theory and Instructional Texts,” Chap. 2, in this book.
- 4.
I will rely on the critical edition given in Peng Hao 彭浩 (2001). It is still difficult to determine the status of this text.
- 5.
- 6.
- 7.
- 8.
In what follows, I refer to any problem from any book by a pair of numbers, the first one indicating the chapter in which the problem was included and the second one giving its rank in the chapter. We will make an exception for the Book of Mathematical Procedures, which was found in separate bamboo strips and the organization of which is still the subject of some debate. Moreover, to allow us to refer to the text of an algorithm in a convenient way, I distinguish steps in the text by introducing letters “(a), (b)…” The reader must keep in mind that these signs do not belong to the original text and are merely a tool for the sake of the analysis. Further, note that I sometimes use letters that do not follow each other alphabetically, in order to allow comparison between the texts of different procedures. Finally, note that in this way, I set up an enumeration that groups together operations that were listed as distinct actions in the sources. More generally, the original texts list operations in a certain way. I group them into larger sets in such a way as to allow comparison between the texts. The beginnings of the groups of operations I form regularly correspond to markers such as “one puts,” “moreover,” “then,” whereas their endings regularly correspond to the assertion of results or to declaratives, giving a name to a value obtained (see below).
- 9.
For want of a better word, throughout this chapter I will use the term “surface” to designate the surface on which one computed in ancient China. Probably, the calculations were not done on a surface specific to that use, but on an ordinary flat area (Martzloff 1987, p. 170).
- 10.
Below, I will systematically represent by columns the configurations of numbers that, during a computation, succeed to each other on the surface, as I suggest recreating them. Moreover, in order not to increase the burden placed on the reader, I replace the counting rods by the Arabic digits that are now common. While each state of the surface corresponds to a column of values, the global computation is represented by the sequence of columns arranged in a table. The reader has to imagine that the columns are states that follow each other, one replacing the other. For the sake of clarity, I divide the columns into zones separated by lines. However, there is no reason to believe that the original surface bore any marks. Let me stress that for the operations discussed, the Mathematical Classic by Zhang Qiujian contained only texts for algorithms, but no further information regarding how the surface was used. In particular, no extant writing composed before the thirteenth century includes illustrations showing the computations on the surface. I do not present any arguments regarding how I recreate the layout of the computations, referring the reader to another publication in which this aspect is addressed in greater detail: Chemla (1996). In particular, note that I interpret “one puts 21” as corresponding to the action of placing 21 on the surface together with the fraction attached to it. Since I am mainly interested in the relationship between the text of the algorithm and the actions to be taken, the way in which the concrete computations are recreated is of no crucial importance here.
- 11.
Most of the articles published about algorithms in ancient China echo this assumption. Let me signal, without dwelling on this issue for the moment, that these assertions are not the only elements of the translated text that do not refer to an event on the surface. The division to be carried out is qualified as being “in return.” This indication points to the motivation for which the division is used here. We come back to such phenomena later.
- 12.
J. Virbel, “Speech Act Theory and Instructional Texts,” this volume, Sect. 2.3, raises this issue by indicating that the aid provided by an instructional text can take “multiple forms” and “be relative” to the user. In Ibid., Sect. 2.3.3.2, he points out that “maximum efficiency,” when the concern is “to give to H (KC, the user of the text) as much aid as possible,” may not be to provide as many details as possible. When Wittgenstein emphasized the import of the surveyability (übersichtlichkeit) of a proof, he pointed out a concern of the same type. I. Hacking quoted and discussed these remarks made by Wittgenstein in his lecture, the text of which can be found at http://www.college-de-france.fr/media/ian-hacking/UPL384701617840960135_4___D__monstration.pdf, pp. 10–11, accessed on April 14, 2015.
- 13.
Let me illustrate this statement by referring to the example of an operation used in present-day mathematical practice, i.e., “reducing to the same denominator.” This operation invites the reader to apply a procedure composed of several multiplications. The text of an algorithm can use either the expression “reduce to the same denominator,” leaving to the reader the task of understanding the actions required, or only give the list of actions. In the latter case, the user possibly remains unaware that the operations fulfill this aim. Another important remark here is that, in the context of ancient Chinese sources, some texts of algorithms introduced operations of the same level as “reducing to the same denominator” to achieve the same result. However, the way in which they group actions to be carried out differs from what “reducing to the same denominator” does. What for us corresponds to “reducing to the same denominator” is prescribed there by means of two higher-level operations, one referring to the action on denominators, the other to the actions on numerators. We thus see how the constitution of such operations may vary according to the context.
- 14.
Here, the competences involve knowing how to lay out the computations on the surface and how to execute the operations. Such competences were necessary for using the written text. Either such knowledge was a prerequisite to reading the text or it was meant to be provided through oral explanations accompanying the text. In any case, the written text did not record any further indications. These questions were outlined in Chemla (2009a). Since we know nothing about the use for which the written text was composed, we cannot go any further here. The question on which we chose to focus, i.e., describing the relationship between the text of the algorithms and the corresponding flows of computation, allows us to bypass this range of consideration in a first phase. However, the conclusions reached could cast light on the issue of the function and use of the writings.
- 15.
We can prove that this is how the commentator Liu Hui, the earliest reader whose reading of a problem can be observed, interpreted a similar text of algorithm in The Nine Chapters, see Chemla (2003). In the case of this passage from the Mathematical Classic by Zhang Qiujian, this interpretation is confirmed by Li Chunfeng’s commentary to be examined below. Note that the reader must also know, in such cases, how to transfer the procedure to a different problem. On this issue compare Volkov (1992), (2008), Chemla (2010b).
- 16.
The Mathematical Classic by Master Sun, one of the ten mathematical classics brought together by Li Chunfeng in the seventh century, provides key information for restoring the arrangement of the values here. In particular, it explains that a multiplicand and multiplier were respectively put in the upper and lower positions of the surface, whereas the dividend and divisor were placed in the middle and lower positions, respectively. These pieces of information guide my way of restoring the computations on the surface here. Compare Chemla (1996). Below, we will see that this is coherent with the information contained in the texts of all the procedures.
- 17.
Compare J. Virbel, “Speech Act Theory and Instructional Texts,” this volume, Sect. 2.3.4.2.
- 18.
In Table 9.2, I suggest interpreting these “positions” as being “upper” and “lower,” respectively. More precisely, when the values placed in a position are integers followed by fractions, these positions become zones in which the quantity is displayed in three rows. Moreover, in my view, computations are carried out in these zones, according to principles, for the layout, that are the same as those described by the Mathematical Classic by Master Sun for the higher level. The term “positions” in the text of the algorithm will designate the value in the central row of each of the zones.
- 19.
Note that what is described in terms of positioning is in affinity with what has been restored above. In particular, these indications confirm that the information found in the Mathematical Classic by Master Sun remains valid in our context.
- 20.
These remarks illustrate one feature of the possible localization of instructional texts—one of the types of difference between such texts introduced by J. Virbel. Compare J. Virbel, “Speech Act Theory and Instructional Texts,” this volume, Sect. 2.3.1, point 10.
- 21.
Like Guo Shuchun 郭書春 and Liu Dun 劉鈍 (2001), p. 343, footnote 4, I consider that the text contained in our sources can be understood and I keep it in the edition of this passage (Chemla 1996). Qian Baocong 錢寶琮 (1963, p. 333), footnote 1, considered it was corrupted and replaced here 下 “below” par 上 “above.”
- 22.
This is the text as given by all documents. Qian Baocong 錢寶琮 (1963, p. 333), footnote 2, considered it was corrupted and replaced it by “之三”. Guo Shuchun 郭書春 and Liu Dun 劉鈍 (2001, p. 343), footnote 5, consider that “三分” is a corruption of the original “三,” hence suppressing 分, which they consider to be an interpolation. If we relied on the evidence provided, for instance, in Chap. 12 of Li Ye 李冶’s Ceyuan haijing 測圓海鏡, we could consider that Qian Baocong was right. However, the text can be understood as it stands. Moreover, the same kind of formulation can be found in the commentary on problem 4.24 in The Nine Chapters (Chemla and Guo Shuchun 2004, p. 378, fn. 375, 534).
- 23.
Our sources contain here “分母三母”. Qian Baocong 錢寶琮 (1963, p. 333), footnote 3 considered that the original text contained here “三分母”. I follow the editorial suggestion made by Guo Shuchun 郭書春 and Liu Dun 劉鈍 (2001, pp. 343–344), footnote 6. However, there are reasons to be discussed below for considering that this point should be revised.
- 24.
Here, and below, my emphasis.
- 25.
More precisely, once the sum of the numerators, 41, and the product of the denominators, 35, were obtained, the procedure could have directly used step (e): multiplying 49 by 35 and incorporating the result to 41. Instead, the procedure given divides by 35, which yields the result of the sum of the fractions. It is followed by an operation canceling its effect: a multiplication by 35. Such procedures are typical of those that the commentators presented to account for why a procedure given in The Nine Chapters was correct. In a first step, they presented the unnecessary operations that brought to light the meaning of the procedure, before, in a second step, deleting them through valid transformations that led to yielding the procedure as found in The Nine Chapters (Chemla 2010a).
- 26.
Table 9.3 shows clearly that the column marked (*) is grey and belongs to the inserted procedure, whereas the column marked (**) is not grey, and belongs to the procedure linked to Problem (1.2).
- 27.
Note that the numbers by which one multiplies (the multipliers) remain on the surface, whereas the values multiplied (the multiplicands) are replaced by the results (see Table 9.3, fifth column). By contrast, the operations formulated in a symmetrical way with respect to their terms correspond to an execution that deletes the initial values and leaves only the result on the surface (probably the result of the sum replaces 21, and the result of the product is placed in the row between the values multiplied (5 and 7)).
- 28.
When Qian Baocong considered here the text as corrupted (see footnote 21), he assumed that positions were designated only in an absolute way.
- 29.
This detail is quite revealing. It indicates that the text is written down in such a way as to indicate by intermittences the meaning of the operations prescribed: the fact that the value computed, 35, is referred to as a “denominator” implies that the result of the computation yielding it is interpreted with respect to the situation, whether this is done by sheer interpretation or by reading the position in which the value is placed on the surface. Moreover, not only is the interpretation achieved, but the outcome of the interpretation of the result is also used in the next step of the text, to refer to the value obtained. Therefore, the user of the text must formulate for himself or herself the meaning of the sequence of results computed in order to be able to use the text. The interpretation is certainly a condition for embedding a procedure into another procedure: the main procedure needs a value for the denominator of the quantity placed in the lower position. Chemla (2010b) reaches the same conclusion from a completely different perspective.
- 30.
See Qian Baocong 錢寶琮 (1963, p. 392).
- 31.
This conforms to the use of the operation “cross-multiplying” in The Nine Chapters, in exactly the same context, see Chemla and Guo (2004, p. 562). In that text too, there is a correlation between the fact that the text makes the positions explicit and the fact that it employs such expressions as “cross-multiplying.”
- 32.
In addition, when the text makes use of the expression “cross-multiplying”, the term discloses a structure in the list of computations, in that two successive multiplications are shown to be linked and parallel to each other. See below the discussion on the property asserted by the introduction of the term “cross-multiplying.”
- 33.
Note that, more fundamentally, a knowledge of the textual operations involved in writing down texts of algorithms, in the culture in which the texts were composed, also appears to have been a competence required for using them.
- 34.
The same holds true for division below.
- 35.
Note that in the texts analyzed above, we indicated how the technical term “cross-multiplying” was used to refer to two related multiplications. Here, the association of computations, which are parallel and related by the fact that they are motivated by similar reasons, is carried out in a different way. In other texts, one can find rather long lists of operations corresponding to such distinct sequences of parallel computations.
- 36.
This last feature characterizes more generally another way of writing down a procedure that can be used for a class of problems. We stressed above the fact that the use of particular values in its text did not in and of itself impair the generality of a procedure. In the case examined here, such a formulation is essential to allow a sentence to correspond to two different and parallel computations.
- 37.
According to the grammar of classical Chinese, the terms “integer,” “numerator” as well as the verb “incorporate” can be interpreted in the singular or the plural. Note that such a choice between the two alternatives, and more generally the interpretation of the text, requires a competence from the user of the text. Again here, such competences are needed for a correct use of the “procedure.” Theoretically, one could also understand the first sentence as follows: “the denominators multiply the integers and (the results) are incorporated into the numerators.” However, if the denominators were different, I believe the sentence would instead be the following: “the denominators respectively multiply the integers corresponding to them and (the results) are incorporated into the numerators.” Also note that the effect of the superfluous multiplication by c is cancelled by a superfluous division by c at the end of the sequence of computations.
- 38.
The reader has understood that I rely on all possible details of the text to restore the corresponding actions. This is the only possible method, if we are to interpret the text in the narrowest context possible. It must be assumed that the practitioner relied on another stock of knowledge to derive actions from the text s/he used to apply for any case encountered.
- 39.
In The Nine Chapters, the procedure given to solve this category of problems is called “Extending generality 大廣.” It differs from the one formulated here by Li Chunfeng in two ways. The formulation is slightly different, in a fashion that is not essential for us here. Moreover, and more importantly, for all the problems, it contains only what corresponds to the second part of Li Chunfeng’s “procedure” here. Li Chunfeng, who also commented on The Nine Chapters, offers an interpretation on the name of the operation. It turns out that this interpretation stresses the unicity of the procedure for all cases, and it is exactly along the same lines as what we suggest here, for the procedure he formulates, that we can understand his own comments on the generality of the procedure “Extending generality” (Chemla and Guo Shuchun 2004, pp. 172–173). More precisely, Li Chunfeng interprets the formulation of the procedure in The Nine Chapters as integrating, in a single text, procedures for solving all possible cases. As a result, if we follow Li Chunfeng’s interpretation of The Nine Chapters, in the way in which the procedure for “Extending generality” is formulated, the first part of the text discussed here is understood as being subsumed under the second part. In other words, the first part is contained in the very formulation of the second part: this is another way of achieving uniformity. This remark hence confirms our interpretation of the procedure Li Chunfeng formulates in his commentary on the Mathematical Classic by Zhang Qiujian: in this procedure, the first part is reshaped in such a way that now a uniform procedure solves all cases. In this commentary, Li Chunfeng has a reason to keep the first and second parts separate, which still has to be made clear. Note also that this type of generality is to be distinguished from the fact that, in contrast to the “detailed procedures,” Li Chunfeng’s “procedure” is described in abstract terms. The interest for uniformity in the treatment of the various cases in ancient China can also be captured in the way in which the commentator Liu Hui deals with problem 6.18 in The Nine Chapters, see Chemla (2003).
- 40.
- 41.
The rewriting or transformation of procedures with the intention of dissolving cases can be evidenced elsewhere in the earliest extant sources from China, see Chemla (2003).
- 42.
Note that the halving is carried out twice to yield the result stated.
- 43.
More precisely, in the Book of Mathematical Procedures the first segment of the procedure—the transformation of the dividend composed of an integer and fraction(s)—is referred to as a change of unit, from which multiplications derive, see Peng Hao 彭浩 (2001, pp. 45–46, 48).
- 44.
This opposition between Liu Hui’s “procedure” and this “detailed procedure” is the same as the one sketched in Sect. 9.1.4 between the formulation in the “detailed procedure” and in Li Chunfeng’s “procedure”. On Liu Hui’s procedure, see Chemla and Guo (2004, pp. 168–169). Its text also has the structure of an enumeration comparable to the one described in Sect. 9.1.4.
- 45.
The first clause of Liu Hui’s procedure deals with the case when a quantity of the type a/b is to be divided by a quantity of the same kind, c/d. It presents common features with another procedure in the Book of Mathematical Procedures for a similar kind of division: “Detaching the length 啟從”. See Peng Hao 彭浩 (2001, pp. 113–115). I will return in a future publication to these various algorithms. Let us stress only that different authors distinguish cases in different ways and thereby shape different general procedures. Moreover, in correlation to the fact that, in his second clause, Liu Hui groups parallel operations in a single sentence, the order in which the computations are to be executed is modified.
- 46.
Probably, this means the upper line in the middle zone of the surface, where the dividend usually lies. In terms of positioning, we follow the same principle as for multiplication, which we derived from the Mathematical Classic by Master Sun.
- 47.
The term “cross-multiplying” in the text corresponds to the performance of a sequence of arithmetical operations on the surface. This occurrence of the term makes clear what the subprocedure includes here. The assertion of the result appears to provide a way of determining the operations covered by the subprocedure.
- 48.
The way of embedding the algorithm for the addition within the algorithm for division is exactly the same here as that described above for the “detailed procedure” following problem (1.3).
- 49.
Or “which is the quantity that multiplies.” One may think that the text is corrupted here and originally contained, instead of “乘數,” “除數,” the whole expression hence meaning the denominator “of the quantity that divides.” However, one may also remark that this editorial problem occurs precisely at the same place, for the text of division as the one discussed above, in footnote 23, for multiplication. The two problems may therefore have to receive a joint solution.
- 50.
The ancient sources have here “三分”. I follow Qian Baocong, when he suggests that the text is corrupted and should be restored as “分母.”
- 51.
It seems to me that the Chinese sentence can be understood here in two ways. Either it lists two types of conditions for which the procedure that follows is to be used: the first condition would correspond to the case when the dividend and the divisor both contain a fraction ( a + b/c and a′+ b′/c′, respectively), the second condition would correspond to the case where there is a sequence of different types of parts attached to either of the two terms of the division ( a + b/c + b′/c′). To express this option, one could emphasize the translation of the conjunction as follows “If the divisor and the dividend both have parts and also in case there are repeated parts.” Or—second option—the text defines here the cases for which the second procedure should be applied by the combination of two conditions: the fact that the dividend and the divisor both have fractions and, furthermore, these fractions have different denominators. It is then translated as follows: “If the divisor and the dividend both have parts and there are repeated parts (i.e., with different denominators).” I opted for the first interpretation in Chemla (1992). The comparison with the first procedure by Li Chunfeng now incites me to opt for the second option. The benefit would be twofold. First, in contrast to what I suggested in my previous article, we would not be compelled to introduce another interpretation for the meaning of the technical expression chong you fen, which would here, as in The Nine Chapters, consistently be understood as “there are different types of parts.” Secondly, the impact on the interpretation of the first part of the procedure would be interesting. We suppose that, as above (Sect. 9.1.4), the cases dealt with in the first part of the text of the procedure are defined by contrast with the condition that marks the beginning of the second part of the text. If we choose the second option, the cases covered by the first part of the procedure would be either of the following: either only one of the dividend and the divisor has a fraction, or they both have fractions, but with the same denominator. Contrary to what the first option would entail, as we will see below, the interpretation based on the second option would make the procedure described here consistent: the second case would in any event be reduced to the first case. Moreover, the computations would be more straightforward. The only problem raised by this interpretation is that it would make the procedure different from that given in The Nine Chapters after a sequence of problems similar to the three problems in the Mathematical Classic by Zhang Qiujian. Let us explain why this constitutes a difficulty, by reminding the reader of the situation of our sources (see Appendix): The Nine Chapters is a key classical text, which Li Chunfeng et al. put together with other classics, including the Mathematical Classic by Zhang Qiujian, when they assembled and annotated the collection of the Ten Mathematical Classics. Why then would Li Chunfeng give a procedure in his commentary on the Mathematical Classic by Zhang Qiujian comparable to, but different from, the one given in The Nine Chapters? The difference could be the following: in the procedure in The Nine Chapters, according to my interpretation (see below), the corresponding step (last case) deals with either the case when the dividend and the divisor have different fractions or one of them has two fractions with different denominators. This would correspond to the first option in the interpretation of Li Chunfeng’s procedure, which, for the reasons explained above, I would rather discard in favor of the second option. Since the first procedure given by Li Chunfeng after problem (1.3) and the formulation of this procedure both suggest that the commentator relies on The Nine Chapters, this difference would require explanation. One possible solution for this puzzle would be to interpret that neither for multiplication, nor for division, does Li Chunfeng’s “procedure” take into account cases when two fractions with different denominators follow each other. In this sense, this also holds true for the alternative procedure offered by Liu Hui in his commentary on The Nine Chapters. We come back to the comparison between the procedure in The Nine Chapters and Li Chunfeng’s below.
- 52.
On this term, the reader is referred to my glossary of terms occurring in the oldest Chinese mathematical texts in Chemla and Guo (2004, pp. 994–998).
- 53.
In each context in which the term is used, “making communicate” has a meaning specific to the context per se—here, making parts share the same size—as well as a formal meaning—“making” entities “communicate.” The term chosen to refer to the operation in our specific context thus indicates the formal meaning. More generally, in the same context, other terms function in the same way. See Chemla (1997b). We will have to come back to this issue from two different perspectives below, in Sect. 9.3. In particular, the second, formal meaning will be best grasped when compared to another way of referring to the same operation.
- 54.
See Chemla and Guo (2004, pp. 342–343).
- 55.
Here again, in fact, in the context within which this sentence occurs, the term can be understood as a singular or a plural, see footnote 4 of the translation into French, p. 798. Since it is not a central issue for us here, I also refer the reader to it for a discussion of the other possible version of the text.
- 56.
We could be tempted to see no difference between adding an integer and a fraction, on the one hand, and adding two fractions, on the other. To understand the difference better, one should recall that in ancient China, an integer was most probably not understood as the numerator of a fraction, whose denominator is 1. Instead, the general type of rational number had the form of a + b/c and, within this context, an integer was approached as a + 0/c. Seen from this perspective, interpreting the operation on the integer by the same term of “making communicate” as the one used, in the previous context, to interpret the operations on the numerators of two fractions establishes a relationship between the two. The use of “making communicate” stresses that, in the motivations, if not in the computations, the two transformations have something formal in common.
- 57.
This is the second kind of indication we have that evidences that readers formulated an interpretation of the effect of the operations at the same time as they turned the sentences into computations—or, to formulate it from the point of view of the authors of procedures: this is the second hint we have that the authors interpreted the meaning of the operations as they wrote them down. If we recall, in footnote 29 above, we stressed that the fact that the text designates “35” as a “denominator” implied that the meaning of the value had already been interpreted. Similarly, here, the fact of designating the multiplication by “making communicate” implies that the intention of the operation achieved is present in the mind of the author of the procedure and must be present in the mind of the practitioner using it.
- 58.
See “to divide in return 報除,” “to divide together 并 除,” “to divide at a stroke 連 除,” in my glossary, in Les Neuf Chapitres.
- 59.
Whether, as in The Nine Chapters, the algorithm covers the case when there can be a sum of two fractions or not (as we explained in footnote 51) is a matter of interpretation. We leave this question aside here. Deciding over this issue does not affect my description of how the term “making communicate” is used. One could develop the same argument on “equalizing.”
- 60.
This was the second option discussed in footnote 51, that is, that the condition refers to the case when the dividend and the divisor both have fractions and these fractions have, in addition, distinct denominators.
- 61.
In addition to the instances discussed below, bringing to light this type of organization for algorithms clarifies the interpretation of several other texts of algorithms in The Nine Chapters, since they share a similar structure. One may think of the algorithms for root extraction (square roots as well as cube roots, in Chap. 4) and the algorithms linked to what was later called in the West ‘rules of false double position’ (Chap. 7). Even the structure of a chapter like fangcheng (Chap. 8) can be better understood from this perspective. The first algorithm it contains solves a specific and fundamental case. Later on in Chap. 8, conditions will be added and the algorithm will be made more general. We cannot discuss these examples further within the scope of this chapter.
- 62.
Let us recall that in case only one term in a multiplication had a denominator, the procedure for different denominators was used, by artificially considering that the integer to be multiplied was followed by a fraction with a zero numerator and the same denominator.
- 63.
It is remarkable that this echoes with the series of problems through which Liu Hui comments on the procedure “Multiplying parts.” Each problem offers a question that an always larger final section of the procedure solves, thereby gradually shaping an interpretation of the final result (Chemla 2009b). I’ll inquire into this parallel in another article.
- 64.
In correlation with this opposition, the text for multiplication presents two parallel and uniform subprocedures before concluding with a common conclusion. In division, however, a complete procedure is formulated, followed by how it is to be modified for the second set of cases. As a consequence, the relationship between the procedures for the first and second sets of cases recalls the relationship between the “detailed procedure” following problems (1.5) and (1.6). In the latter case, a procedure, that for adding fractions, was grafted onto another procedure, reducing the more complex case to a simpler one. The text of the algorithm for division is built according to the same logic.
- 65.
- 66.
When Li Chunfeng comments on the procedure for multiplying quantities which is called “Extending generality,” as evoked above (See footnote 39), he interprets the text as incorporating three cases (multiplying two integers, multiplying two fractions, multiplying two integers followed by fractions). However, there too, there is no problem in the immediate context which raises the case of multiplying two integers.
- 67.
Some slight differences can be noted in this respect. In Li Chunfeng’s text, the operations prescribed in the second item overlapped with the beginning of the operations listed in the first item. However, the procedure for the cases covered by the second item still had to be completed with the last operation prescribed in the first item. In the text for “Directly sharing” included in The Nine Chapters, the operation for the second item is part of those given for the third, whereas the two types of procedures are concluded by the operations placed at the beginning of the text.
- 68.
I will not enter into all details, for this the reader is referred to the critical edition, the French translation, and especially to the footnotes and the glossary through which I comment on the procedure and the commentaries (see Chemla and Guo Shuchun 2004, respectively, pp. 166–169, 766–767, and 993–998).
- 69.
I suggest here departing from what was suggested in footnote 18, p. 158, in our critical edition and, rather, follow Dai Zhen (after 1776, see Guo Shuchun in Chemla and Guo 2004, pp. 158, fn. 18), Qian Baocong (1963) and Li Jimin (1993, p. 162, fn. 26), in considering that “知” that is the reading in all ancient sources is a corruption of “者.”
- 70.
In my glossary, I accounted for this phenomenon by suggesting that the terminology may have changed between the time of The Nine Chapters and that of Liu Hui. However, the previous analysis seems to reveal that it may have been common to use a single term to refer to different procedures, provided that they all shared the key operation referred to by the term. Therefore, we may also have here another occurrence of the same general phenomenon.
- 71.
This echo seems to indicate that there is a correlation between how Liu Hui develops proofs for the correctness of procedures and what he reads in the Classic. Moreover, he seems to read The Nine Chapters as possibly referring to reasons of the correctness by the way in which directives are carried out. We come back to this key issue below. It is developed in greater detail in Chemla (1992, 1997b, 2010a).
- 72.
In the first segment of Li Chunfeng’s general procedure for division analyzed in the previous section, this operation was not prescribed in relation to “making” entities “communicate.”
- 73.
The term of lü qualifies quantities that represent the values of magnitudes only relatively to each other. These quantities can hence be multiplied or divided by the same amount, without damaging the property of the quantities obtained to also represent the relationship between the magnitudes. On this term, see my glossary in Chemla and Guo (2004, pp. 956–959). I refer the reader to Chemla (2010a, p. 274) for a more detailed analysis of the second layer of meaning that Liu Hui reads here in “making communicate.”
- 74.
As in footnote 69, I follow Dai Zhen’s idea, after 1776, in thinking that “知,” which the ancient sources contain here, is a corruption of “者.”
- 75.
The same term lü is used here in a verbal sense. It also occurs with the same meaning and function in The Nine Chapters, in a similar set of problems, which is placed in the second half of Chap. 2. I have already dealt with this extremely interesting procedure from different angles (its connection to procedures in The Nine Chapters, the connection between the formulation of procedures and the proofs of their correctness, the prescription by analogy) in, respectively, Chemla (2006, 2010a, 2010b). I refer the reader to these other publications for a more detailed treatment which I do not repeat here. In particular, I do not comment on the editorial problems raised by the passages considered, nor do I account for how I interpret it by contrast to how other historians interpret it. Since my 2006 article was published, two new books on the Book of Mathematical Procedures have appeared: Horng Wann-sheng洪萬生, Lin Cangyi林倉億, Su Huiyu蘇惠玉 and Su Junhong蘇俊鴻 2006, 張家山漢簡『算數書』研究会編 Chôka zan kankan Sansûsho kenkyûkai. Research group on the Han bamboo strips from Zhangjiashan Book of Mathematical Procedures 2006.
- 76.
See Peng Hao 彭浩 (2001, pp. 73–74). The reader should keep in mind the fact that the book was found as separate bamboo strips, while they were originally bound together in a roll. Each edition hence suggests different orders for arranging the strips. For instance, in the Japanese edition mentioned in the previous footnote, the sections we are interested in are placed as the 47th and the 48th. I systematically refer the reader to the numbers attached to the bamboo strips as provided by Peng Hao 彭浩 (2001) and I follow, for the time being, the organization of the book as suggested in this publication.
- 77.
These are three units of capacity, and the relationships between them are: 1 shi 石 = 10 dou斗 = 100 Sheng 升.
- 78.
Here, to be more precise, the sentence “One also triples the quantity of sheng of 1 shi 亦三一石之升數” prescribes two operations. Transforming “1 shi” into sheng is prescribed by the statement of the result (“the quantity of sheng of 1 shi 一石之升數”). Such prescriptions are frequent in the texts from ancient China. This first one is combined with the prescription of tripling.
- 79.
I have dealt with the history of the use of the term “also” in the texts of procedures known from ancient China in Chemla (2010b).
- 80.
There are fundamental reasons for this, which I examined in Chemla (2006). The formulation that the text for “Directly lü-ing” receives in The Nine Chapters is quite interesting too. However, its analysis exceeds the scope of this chapter.
- 81.
I cannot develop this point here. I refer to the publication in which I will analyze the system of such procedures that the Book of Mathematical Procedures contains.
- 82.
This corresponds to the initial item of the procedure for “Directly sharing” in The Nine Chapters.
- 83.
Here, the actual configuration of numbers on the surface is indicated in the middle column. To help the reader follow, I add words in the middle column, as well as in the left and right columns. However, there is no reason to believe that anything was written on the surface.
- 84.
By “similar,” I mean any highest unit in a scale. In fact, in the Book of Mathematical Procedures, other similar procedures are given for measuring units having other positions in a given scale. This kind of problem plays a key role in the book.
Bibliography
Chemla, K. 1992. Les fractions comme modèle formel en Chine ancienne. In Histoire de fractions, fractions d’histoire, eds. P. Benoit, K. Chemla and J. Ritter, 189–207, 405, 410. Basel: Birkhäuser.
Chemla, K. 1996. Positions et changements en mathématiques à partir de textes chinois des dynasties Han à Song- Yuan. Quelques remarques. In Disposer pour dire, placer pour penser, situer pour agir. Pratiques de la position en Chine, eds. K. Chemla and M. Lackner, 115–147, 190, 192. Saint-Denis: Presses Universitaires Vincennes.
Chemla, K. 1997a. Reflections on the world-wide history of the rule of false double position, or: How a loop was closed. Centaurus. International Journal of the History of Mathematics Science and Technology 39: 97–120.
Chemla, K. 1997b. What is at stake in mathematical proofs from third-century China? Science in Context 10: 227–251.
Chemla, K. 2003. Generality above abstraction: The general expressed in terms of the paradigmatic in mathematics in ancient China. Science in Context 16: 413–458.
Chemla, K. 2006. Documenting a process of abstraction in the mathematics of ancient China. In Studies in Chinese language and culture—Festschrift in Honor of Christoph Harbsmeier on the Occasion of his 60th Birthday, eds. C. Anderl and H. Eifring, 169–194. Oslo: Hermes Academic Publishing and Bookshop A/S. http://halshs.archives-ouvertes.fr/halshs-00133034, http://www.instphi.org/Festschrift.html.
Chemla, K. 2009a. Literacy and the history of science. Reflections based on Chinese and other sources. In Cambridge handbook of literacy, eds. D. R. Olson and N. Torrance, 253–270. Cambridge: Cambridge University Press.
Chemla, K. 2009b. On mathematical problems as historically determined artifacts. Reflections inspired by sources from ancient China. Historia Mathematica 36: 213–246.
Chemla, K. 2010a. Proof in the wording: Two modalities from ancient Chinese algorithms. In Explanation and proof in mathematics: Philosophical and educational perspectives, eds. G. Hanna, H. N. Jahnke and H. Pulte, 253–285. Dordrecht: Springer.
Chemla, K. 2010b. Usage of the terms “likewise” and “like” in texts for algorithms. Algorithmic analogies in ancient China. In Analogien in Naturwissenschaft und Medizin, ed. K. Hentschel, 329–357. Halle: Leopoldina (Acta Historica Leopoldina).
Chemla, K. 2012. Reading proofs in Chinese commentaries: Algebraic proofs in an algorithmic context. In The history of mathematical proof in ancient traditions, ed. K. Chemla. Cambridge: Cambridge University Press.
Chemla, K., and Guo Shuchun. 2004. Les neuf chapitres. Le Classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.
Guo, Shuchun 郭書春, and Liu Dun 劉鈍. 2001. Suanjing shishu 算經十書. Guo Shuchun, Liu Dun dianjiao 郭書春,劉鈍 點校. Chonghe 中和 (Taipei county 台北縣): 九章出版社 Jiu zhang chubanshe.
Horng, Wann-sheng 洪萬生, Lin Cangyi 林倉億, Su Huiyu 蘇惠玉, and Su Junhong 蘇俊鴻. 2006. Shu zhi qiyuan 數之起源 (The origin of mathematics). Taipei 台北: Taiwan Shangwu yinshuguan 臺灣商務印書館.
Høyrup, J. 1990. Algebra and naive geometry: An investigation of some basic aspects of Old Babylonian mathematical thought. Altorientalische Forschungen 17: 27–69, 262–324.
Høyrup, J. 2002. Lengths, widths, surfaces: A portrait of old Babylonian algebra and its kin. Sources and studies in the history of mathematics and physical sciences. New York: Springer.
Li, Jimin 李繼閔. 1993. Jiuzhang suanshu jiaozheng 九章算術校證 (Critical edition of The Nine Chapters on Mathematical Procedures). Xi’an: Shaanxi kexue jishu chubanshe.
Martzloff, J.-C. 1987. Histoire des mathématiques chinoises. Trans. S. S. Wilson. Paris: Masson.
Peng, Hao 彭浩. 2001. Zhangjiashan hanjian «Suanshushu» zhushi 張家山漢簡《算數書》注釋 (Commentary on the Book of mathematical procedures, a writing on bamboo strips dating from the Han and discovered at Zhangjiashan). Beijing: Science Press (Kexue chubanshe).
Qian, Baocong 錢寶琮. 1963. Suanjing shishu 算經十書 (Qian Baocong jiaodian 錢寶琮校點) (Critical punctuated edition of The Ten Classics of Mathematics). 2 vols. Beijing 北京: Zhonghua shuju 中華書局.
Volkov, A. 1992. Analogical reasoning in ancient China: Some examples. In Regards obliques sur l’argumentation en Chine. Extrême-Orient, Extrême-Occident. 14, ed. K. Chemla, 15–48. Saint-Denis: Presses Universitaires de Vincennes.
Volkov, A. 2008. Raisonnement par analogie dans les mathématiques chinoises du premier millénaire de notre ère. In L’analogie dans la démarche scientifique. Perspective historique, ed. M.-J. Durand-Richard, 61–95. Paris: L’Harmattan.
張家山漢簡『算數書』研究会 Chôka zan kankan Sansûsho kenkyûkai. Research group on the Han bamboo strips from Zhangjiashan Book of Mathematical Procedures. 2006. 漢簡『算數書』Kankan Sansûsho. The Han bamboo strips from Zhangjiashan Book of Mathematical Procedures. 京都 Kyoto: 朋友書店 Hôyû shoten.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix: The Sources Used in this Chapter
Appendix: The Sources Used in this Chapter
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Chemla (林力娜), K. (2015). Describing Texts for Algorithms: How They Prescribe Operations and Integrate Cases. Reflections Based on Ancient Chinese Mathematical Sources. In: Chemla, K., Virbel, J. (eds) Texts, Textual Acts and the History of Science. Archimedes, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-319-16444-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-16444-1_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16443-4
Online ISBN: 978-3-319-16444-1
eBook Packages: Humanities, Social Sciences and LawHistory (R0)