Zhangjiashan Han Jian's "Book of Arithmetic" has three arithmetic problems related to "Zhu Cai", "Yi Zhu Cai", and "Yi Fang Cai". Because there are too many indistinguishable places in the original handwriting, there are many missing texts in the original interpretation, and there is no satisfactory interpretation so far, while the two questions of "using the square material" and "using the square material" are inconsistent with the simple mathematical formula, resulting in different opinions and inconsistencies among scholars. In this paper, we first deal with the indistinguishable parts of the original and simplified handwriting of "Zhucai", obtain key clues from it, and verify the correct interpretation. For the two arithmetic problems of "using the square material" and "using the square material", we will carry out the argumentation according to the level of ancient mathematics, the practical nature of the application of the arithmetic problems, and the examples in the "arithmetic book", so as to obtain a correct interpretation.
Zhangjiashan Han Jian's "Arithmetic Book" contains a total of two Jian, Zhangjiashan No. 247 Han Tomb Bamboo Slip Finishing Group's interpretation is:
睘 (圜) material There is a circle material (?Broken in the city, the city, the big geometrySaid: Seven (?.)Ten (?.)Six (?.)Four and a half inches. Narrative (technique) said: 囗Self-multiplication, to (156).
into two inches of benefit, that is, a large number has been. （158）
Since many of the texts cannot be interpreted, none of them can read through this question. The author has carried out a certain degree of technical processing on the original plate, so that some keywords can be read, and the results of some plate processing are as follows:
As can be seen from Figure 1, there is still room after "圜材", and there are ink stains, so the word "one" in the original interpretation is preferable. As can be seen from Figure 2, the original interpretation of the text "Broken City" is problematic, and it can be found that the first character of the bamboo slip in this paragraph has two small "mouth" characters on the left, which are obviously larger than the small "mouth" of "斷", and there is no ink blot with so many strokes of the word "斷" under it, and there is no residual ink in the shape of the word "斷" on the far left, and the space is not enough to accommodate the lower left part of the word "斷". Therefore, this character is not a "broken" word, but should be a "斲" character. Based on the handwriting and other simplified texts in Figure 2, it can be found that this character is indeed the word "斲", and this paragraph of the simplified text is "斲之入二區".
The difference between the words "斲" and "broken" is related to the understanding and interpretation of the whole question. The book of Qin Jian's "Numbers" and the "Nine Chapters of Arithmetic Pythagorean" in the collection of Yuelu Academy both have the arithmetic problem of "cutting materials", which are quoted as follows:
Today] there is a round timber [buried] ground, unwise [know] small and large, cut it, into the material an inch and get a flat foot, ask how big is the material (can) [what]?That is: half flat to get five inches, so that multiplication also, to a deep inch for the law, such as the law to get an inch. There is [also] to benefit deeply, that is, the diameter of the material.
Now there are round timbers, buried in the wall, and the size is unknown. With a saw, it is one inch deep, and the saw path is one foot long. Q: Diameter geometry?Answer: The diameter of the material is two feet six inches. The technique says: Half sawing road multiplication, such as the depth of the inch and one, to the depth of the increase of the inch, that is, the diameter of the material.
Zhangjiashan Han Jian's "Arithmetic Book" is a "round material", and the simplified text has the sentence "cut into two inches", and "and get", "ask the big geometry", "inch self-multiplication", and "into two inches of benefit" can be clearly interpreted, and they are all in a suitable position in the simplified text. Therefore, compared with the arithmetic problems in Yuelu Qin Jian's "Numbers" and "Nine Chapters of Arithmetic", this problem is undoubtedly a similar "cutting materials" arithmetic. Therefore, the words that cannot be confirmed in the arithmetic can be judged with the help of the way the two arithmetic questions are written.
The first character of the bamboo slip fragment shown in Figure 3 is illegible, the position of the second character should be "ruler" according to the context and the shape of the ink, the third character is a number according to the meaning of the text, and the last word should be "six" or "four" according to the shape of the ink, and the last word can be affirmed as "inch" according to the meaning of the text and the residual ink. This bamboo slip is preceded by the word "and de", in which the word "and" is recognizable, and the word "de" can be determined according to the context. According to the corresponding content of the Qin Jian's "Number" book in Yuelu Academy, it can be judged that these four characters are "four inches of flat feet" or "six inches of flat feet".
In Figure 4 and Figure 5, the original interpretation reads "曰七(?Ten (?.)Six (?.)and "four and a half inches". According to Figure 5, the word "inch and a half inch" is recognizable, while the first word is judged by the residual ink, or the word "four" or "six" in the original interpretation. Figure 4 is illegible except for the word "曰", but it is known according to the context that it is "曰大囗ruler囗". Considering that the simplified text in Figure 5 is "囗寸半珏", and the traces of the residual ink after the word "ruler" are similar to "有", the word should be "有[and]" according to the text. So far, according to the calculation, this answer is "two feet have six and a half inches", and Figure 3 is "four inches flat".
According to the text of Yuelu Qinjian's "technique" quoted above, it can be seen that No. 158 Jane is missing "as the law, and into two inches" seven words. Because "into two inches" appears twice in our restored simplified text, the reason for this error is obvious: it is because there are two places "into two inches" above and below the simplified text, and the copyist accidentally involves each other and produces a mistake. So far, we have obtained the following brief text of the restoration:
The eyelid material has a round of wood, cut into two inches, and get a square foot of four inches, asking how big the material isHe said: Two feet big is six and a half inches. Narrative [technique] said: Seven inches multiplied by (156).
Entering two inches [for the law, and into two inches] is beneficial, that is, a large number has been. （158）
Here, the "large" in 158 is the "large material" in the question, and the "large number" is the size of the diameter of the round material. The Nine Chapters of Arithmetic classifies such arithmetic problems into the chapter "Pythagorean", and its calculation formula is derived from the Pythagorean theorem, and the derivation process is as follows:
As shown in the figure above, (radius-depth), semi-flat, and radius form a right-angled triangle, according to the Pythagorean theorem
Radius-Depth) (Radius-Depth) Semi-Flat Semi-Flat Radius Radius.
Expand, simplify, get:
2 deep radius half flat half flat deep deep.
Therefore, the diameter is half flat and half flat and deep and deep.
The descriptions of the algorithms of this question, Yuelu Qin Jian, and "Nine Chapters of Arithmetic" are all different from each other, but they are all the above formulas. According to this formula, this question is calculated as follows:
The diameter of the round timber.
Zhangjiashan Han Jian's "Arithmetic Book" has always been regarded by researchers as either wrong or wrong in the data because the formulas provided by the "technique" are different from the so-called correct formulas derived from mathematics. However, this is not the case, and we will prove that the formulas in these two calculations are not purely mathematical formulas, but they are applied formulas for measuring the size of square timber made of round timber according to the actual production situation. This subsection first discusses the arithmetic problem of "using the material to solve the problem", and its brief text is as follows:
Take the square of the wood to the round wood as the square material, said: the big four Wei [circumference] two inches twenty-five minutes and fourteen, for the square material geometrySaid: Seven inches and five minutes and three inches. Shu said: Therefore five is true, so that seven and one, four (153).
And] one is done. （157）
Su Yiwen et al., Duan Yaoyong and Zou Dahai, as well as experts from the Zhangjiashan "Arithmetic Book" Research Association in Japan, believe that the word "and one" should be added. This kind of proofreading syntax is smooth and consistent with the number of answers, and the meaning of the text is complete and reasonable. However, according to the plate, the bamboo slip of this title is No. 153 Jane, and its simple text to the "four" character has reached the lower end of the bamboo slip to weave the line, which shows that the "technique" here is not a lack of text, but a lack of a follow-up bamboo slip. Jane No. 153 is numbered H103, while Jane No. 157 is numbered H102, and the two Jane excavation numbers are connected, and the unearthed locations are indeed adjacent. Since the entire text of Brief No. 157 is "ready-to-do", the content is connected to the content of Brief No. 153. As discussed in the previous section, Jane No. 157 does not belong to the topic of "Zhucai", so we determine that Jane No. 157 is a continuation of No. 153, and the word "and" is mistakenly omitted between the two Jane texts. Therefore, we have listed the number 157 after this summary and corrected the word "[and]" with the example of detachment.
In the simplified text, "Wei" is borrowed as "circumference", and in ancient times, "diameter ruler is circumference", and "circumference" here is actually equal to "ruler". Therefore, "the circumference of the big four inches is 2 inches, 25 minutes and 14 inches" means "round wood", and the circumference is inches. According to the calculation of the "Shu" text, the side length of the "square material" is: , which is consistent with the answer. Therefore, we believe that there is no problem with the algorithm and the brief text of the answer. Peng Hao, Guo Shirong, and Guo Shuchun believe that this question is wrong, and they have interpreted and corrected it with their own intentions, and we are the same as Duan Yaoyong and Zou Dahai, and we think that they are not correct.
Although there are a certain number of errors in all the 60 arithmetic problems in the "Arithmetic Book" before this problem, only one arithmetic problem is "women's weaving" that produces wrong answers due to wrong "techniques", and the "women's weaving" problems are "interesting mathematics" rather than "practical mathematics" in nature. In other words, the algorithms for the practical arithmetic problems in the "Arithmetic Book" are correct. Therefore, from a statistical point of view, it is very unlikely that the "technique" of this question is wrong. The algorithm of the "technique" of this question is consistent with the answer, and corresponds to the "four and one" of this question, and the algorithm in the "square material" question discussed later is also correspondingly "thus four". It can be seen that there is definitely no problem with the "technical" text and the answer to this question.
The ancients "used a gauge for a circle, and a moment for a square", so the relationship between the square and the circle within the circle was obviously clear to the ancients, and the above quotation of "Zhucai" proved that the Pythagorean theorem was also a well-known knowledge in the Qin Dynasty, and from the "Fang Tian" problem of Zhangjiashan Hanjian's "Book of Arithmetic" and the "Nine Chapters of Arithmetic" it can be seen that the ancients often used approximate formulas when opening squares for natural numbers of shapes.
According to this formula, then, therefore, the side length of the hypotenuse of a square with a side length of 5 is slightly greater than 7, which shows that the so-called "seeing evil and seeking a prescription, five, seven and one" in the "Sun Tzu Sutra" is an approximate formula that has existed in ancient times. In addition, from the arithmetic problems such as "Cover Cover", "Pavilion Pavilion" and "Well Material" in Zhangjiashan Hanjian's "Book of Calculations", it can be seen that "the diameter is three times a week", that is, the pi is about equal to three, which was also a well-known thing at that time. After an in-depth study of the hand-me-down documents and Qin Jian, Mr. Zou Dahai came to the conclusion that "the main algorithm of the Nine Chapters had already been used in the pre-Qin period" and that "the pre-Qin period must have used the high-level mathematical knowledge achieved in the Han Dynasty and the Nine Chapters", and these conclusions also support our view from a macro perspective. Therefore, we can be sure: and in the Qin Dynasty is a well-known fact. Therefore, if you apply "see the evil and seek the prescription, five, seven and one", then the formula of "taking the circle material as the square material" should be:
Circumference 5 7 pi.
If you replace the pi in the above formula with "three days of diameter", the formula becomes:
Circumference of the wood. In other words, it seems that the last sentence of the "technique" essay in this question should be "three and one" rather than "four and one". However, we have already shown that the algorithm of this problem is consistent with the number of answers given by the arithmetic problem, so it is impossible to make mistakes, so the "four and one" of the "Shu" text does not originate from wrong mathematical knowledge, but has another reason.
The fact that pi is slightly greater than three is a fact that the ancients knew. If the algorithm is "three and one", then the divisor is small, the side length of the calculated inscribed square will be larger than the actual possible size, plus the approximation of "five, seven and one", it can be seen that the answer calculated according to the "diameter one week three" will be 6% larger than the actual number, it can be seen that in practical application, the side length of the "square wood" that may be obtained from the "circle wood" cannot be calculated with "five, seven and one" and "three and one", that is to say, in terms of practical problems, "three and one" is not feasible. On the other hand, this problem is a practical problem, the actual "round timber" may not be so round, its tail diameter must also be slightly smaller, and even the surface layer of the wood may not be able to be used, so when calculating the diameter of the "square timber" that can be obtained, the estimation of the margin is the need of the practical problem. The brief text of this arithmetic problem clearly says that "the round material is the square material", which is extremely worthy of attention!This brief text explains that the abbreviated title of "taking the circle as the square" is only an abbreviation of the name of the problem, and the problem is talking about "circle" and "square", not "circle" and "square", that is, this problem is a practical problem rather than a pure mathematical problem. As discussed above, it is out of practice to use "three and one" to calculate the side length of square timber, which shows that the calculation formula of "four and one" is the method of estimating the edge length with margin, rather than a mathematical error. Experts have not seen this before, and inappropriately equate this problem with such a simple mathematical problem as "knowing the circumference of a circle, and finding the length of the sides of its inner square", which is why they think that this problem is wrong.
All in all, the ancients knew that pi was about 3, but they used "four and one" in the calculation problem, and the calculation formula is completely consistent with the answer, so it can be seen that the "four and one" is an estimation formula with a margin given because it is an application problem of "using the circle as the square material". Although the experts of the Zhangjiashan "Arithmetic Book" Research Association in Japan did not demonstrate in detail, they also believed that "four and one" was a calculation method with a margin left in practice. In addition, from the various excavated Qin slips, it can be found that the Qin people practiced strict and highly quantitative management at the grassroots level, so we further believe that the calculation method of "four and one" in the problem of "using the circle as the square material" is not arbitrary, and should be based on the official regulations equivalent to "cheng".
In the proofreading of the "Method" arithmetic problem, the experts mainly think that there is a mistake or plagiarism in the calculation formula, or that the data in the arithmetic problem is wrong, in short, the error is not serious to a certain extent. The "square-based" arithmetic problem is different, and many experts believe that the algorithm of this arithmetic problem is fundamentally wrong. These experts believe that the algorithm of this problem is a fundamental mistake in treating "using square materials" as the inverse problem of "using square materials", but in fact they misunderstand this problem. Let's take a look at the brief text of this arithmetic first:
To square timber to square for the circle, said: the material square seven inches five minutes of the three, for the circle of material geometrySaid: Siwei [Wai] two inches twenty-five minutes fourteen. The technique says: One side of the square material is (154).
The diameter of the material is also, so the four, [seven] is true, so that the five become one. （155）
In the simplified text, the word "said material" was changed by Duan Yaoyong and Zou Dahaixiao to "material said", and "material" belonged to the previous sentence to form the word "圜材". We believe that in comparison with the following, it can be seen that the word "material" here is "square material", and because there is the word "square" after it, the province of "square tim" is called "material", so it is appropriate to retain the original simplified text. In addition, the sentence "one side of the square material is the path of the round wood" in the text of this question "technique" seems to be difficult to read. However, although the meaning of "diameter" is similar to the original meaning of "diameter", it should be understood as "cross-section" according to the context. Therefore, we believe that this sentence is to explain the relationship between the "square timber one face" and the "round timber" cross-section.
Comparing the data, answers and the text of the algorithm in the known conditions of the problem, this question is indeed the inverse problem of "using the square material" as the "using the square material" problem, and according to the algorithm of the "using the square material" question, it can be seen that the original and simple words "seven" are mistakenly removed, so the correction is as above. According to the algorithm after proofreading, the specific calculation of this problem is: the diameter of the material is calculated by the data in the known conditions according to the algorithm and the result is completely consistent with the answer in the calculation problem.
As mentioned above, this problem is an inverse problem of the "See-to-See" problem from the perspective of the algorithm. As we have already demonstrated in the previous section, the "square with the wood" is a practical problem rather than a simple mathematical problem, where the "four and one" algorithm is derived from the machining allowance for practical operability. In other words, the problem of "square with a round timber" is an application problem of "finding the edge length of the square timber obtained from the circumference of the round timber is known". This problem is not only an inverse problem of "using the square of the square timber", but also a practical arithmetic problem of "knowing the side length of the obtained square timber and finding the circumference of the required round timber".
However, since domestic experts believe that the problem of "using the square of the square" is a pure mathematical problem of "knowing the circumference of a circle and finding the length of the side of the square with its inscribed square", this problem is accordingly understood as the problem of "knowing the side length of the square and finding the circumference of its inscribed circle". Since such a problem is not an inverse problem of "using the material to solve the problem", experts such as Peng Hao, Guo Shirong, Guo Shuchun, Duan Yaoyong and Zou Dahai all believe that the algorithm of this problem is wrong, and give their own different correction plans.
We believe that it is a hypothesis to assume that the Book of Arithmetic will have an algorithmic error in such a practical application problem. Judging from the entire "Arithmetic Book", it is a reference book for low-level officials in the Qin and Han dynasties when they encountered mathematical problems in actual management, and their problems are basically all real-life problems. As far as the problem of "square timber is concerned", the practice of leaving a margin when calculating the square timber cut from round timber is obviously the need of the actual arithmetic problem. Therefore, if the algorithm of this problem is wrong, it should have been corrected by practice. In addition, the original meaning of "wood" and the most commonly used word meaning are "wood", in terms of wood, the processing of "round wood" into "square wood" is a common thing in real life, on the contrary, ancient logs are very easy to obtain in the official, and how many officials need to process "square wood" into "round wood".Therefore, as a practical arithmetic problem, this problem can only be an application problem that officials need in their actual management: a certain project needs a certain size of square timber, so what specifications of round timber should be issued to craftsmen by the official?
Since this question is the inverse problem of the problem of "using the square of the wood", then the question should be "the circumference of the circumscribed circle of the square (on the premise of leaving a margin)", and the domestic experts believe that this question is about the "inscribed circle" rather than the "circumscribed circle", in addition to not understanding the practice of "four and one" in the question of "using the square of the material" to leave a margin, but also because they understand the word "for" in the question of "geometry for the material" in this question. Based on the understanding of the text alone, the word "for" in the sentence "taking square timber as a round timber" seems to be taught to "make", so the text is naturally interpreted as "obtaining inscribed round timber from square timber". However, this understanding lacks a comprehensive analysis of this topic and this sentence in the light of reality, which contradicts our above argument and is therefore unreliable. The word "for" has many meanings since ancient times, and it may not be taught "to do" here.
In fact, in addition to training "make", the word "for" in the ancient text can still be trained to "use", "say", "say", "in", "such as", etc., therefore, considering our above discussion, the word "for" in the question of "for the geometry of the material" in this question does not need to be trained as "made", and its meaning may be close to "use" and "need". An example of an explanation of the word "for" can be found in Zhangjiashan Hanjian's "Book of Arithmetic". "Cheng He" inscription in the "Book of Arithmetic" says that "one stone of He millet is sixteen buckets of millet and half a bucket, and the inscription of "rice seeking su" says "there are seven points of rice and six liters of rice, which should be the geometry of millet today", although these two paragraphs of text are the conversion of "millet" and "rice", but the meaning and usage of the word "for" are different. When the inscription of "Cheng He" says "for the rice and a stone", it is clearly said that "舂之", so the word "for" is the same as the word "for" in the sentence of "to take the material of the square" in the title of "to take the material of the square". However, "millet" can be used as "rice", but "rice" cannot be made into "millet", so in the title of the "rice for corn" arithmetic problem and its sentence "when it is millet geometry", the meaning of the word "for" is close to "need", but in fact, it can be simply interpreted as "conversion". The use of the word "for" in the sentence "when it is millet geometry" is comparable to the sentence "for the material geometry". It can be seen that the word "wei" in the two arithmetic problems can be interpreted differently, and the situation is similar to the conversion problem of "millet" and "rice". In fact, the arithmetic problems between "millet" and "rice" in the Qin and Han dynasties can all be regarded as proportional conversion, and similarly, the two problems of "using the square material" and "using the square material" can also be regarded as the "conversion" problem between the "round material" and the "square material".
In short, based on our arguments in the "Prescription with Radiant Material" and "Prescription with Radiant Material" problem, the conclusion is very clear: the "Prescription Material" problem is a practical problem that calculates the size of the required "Circle Material" according to the size of the target "Prescription", and it is the inverse calculation problem of the "Prescription with Radiant Material" problem, and the question, algorithm and answer of this problem are not wrong except for the mistake of the word "Seven". The algorithms for the two arithmetic problems are correct, and they are practical application formulas based on the allowance rule.