Study on Compilation of Bi Li Shu Biao

Bi Li Shu Biao is a logarithm table compiled by Jean Nicolas Smogolenski and Xue Fengzuo in the beginning of Qing dynasty and it has ten thousand common logarithm values of natural number. By analyzing these values, we found that they were obtained through cutting western logarithm values and adopting the method of five homes six into on the basis of reference to the Western logarithmic table. Almost logarithm values is correct, its false rate is only 0.76 percent.


Introduction
Bi Li Shu Biao is a common logarithm table compiled by Jean Nicolas Smogolenski and Xue Fengzuo in the beginning of Qing dynasty. With this table, they introduced the western logarithm to China, and improved the development of Chinese mathematics and astronomy. Qian Baocong ever said (Qian, 1964), "It (the logarithm) is very useful, and it was used in Chinese astronomy and Calendar immediately at that time." So many researchers have studied this table, such as Mei Wending, Ruan Yuan, Li Yan and so on. Through studying this table even they have got many results, the question about how the table is compiled at that time does not have answer yet. However, it is necessary to find the answer, since it not only can improve our understanding to this table, but also help us know the works of Jean Nicolas Smogolenski and Xue Fengzuo well. So we are going to study it.

Content of Bi Li Shu Biao
Bi Li Shu Biao had definitely only one volume, even though it marked with twelve volumes on its first page (Guo, 2011). It included ten thousand common logarithm values of natural numbers which were from 1 to 10,000. The way that the numbers was arrayed was the natural numbers are on the left and their logarithm values were on the right. So indeed the table totally had twenty thousand numbers. This table is as shown Figure 1.
Each logarithm values of natural number in this  (Graham, 2003), such as John Napier, Briggs Henry and Adrien Vlacq (1600-1667) and so on. Two logarithm tables created by Briggs Henry and Adrien Vlacq respectively in 1624 and in 1628 are as shown in Figure 2 and Figure 3 (Herny, 1624& Adrien, 1628.

Obtaining of Logarithm Values in Bi Li Shu Biao
Jean Nicolas Smogolenski ever said in the beginning of the table, "The old book written long time ago had one hundred thousand numbers, unfortunately I lost most of them on the way, only ten thousand numbers left." (Han 2007). So the logarithm values in Bi Li Shu Biao must be obtained on the basis of reference to the Western logarithmic table and by processing.
Then how did Jean Nicolas Smogolenski and Xue Fengzuo process? How many works did they do?

Obtaining of Logarithm Values in Bi Li Shu Biao
Jean Nicolas Smogolenski ever said in the beginning of the table, "The old book written long time ago had one hundred thousand numbers, unfortunately I lost most of them on the way, only ten thousand numbers left." (Han 2007). So the logarithm values in Bi Li Shu Biao must be obtained on the basis of reference to the Western logarithmic table and by processing. Then how did Jean Nicolas Smogolenski and Xue Fengzuo process? How many works did they do?
To find the answer, we calculated the all ten thousand logarithms values with the formula 6 10 lg A N = by modern computer. The results are shown as Table 1.
Comparing Table 1 and Bi Li Shu Biao, we found there are over thousand logarithm values of natural numbers are different. Such as the logarithm value of 231, the logarithm value of 274 and the logarithm value of 312 and so on. These new logarithm values usually are 1 bigger than old logarithm values in Bi Li Shu Biao.
What happened? Is it the method of four homes five into led to those differences? Since our default setting about carry method is four homes five into. To find the reason, we calculate all ten thousand logarithm values again. We arranged all logarithm values must be reserved three digits after decimal point at this time. New results are shown as Table 2.  From Table 2, we know that the first digit after decimal point of those different logarithm values which got after first calculation all are 5. So the logarithm values which first digit after the decimal point is 5 seems not be carried, that is all 5 that is after decimal point had been canceled. Is the carry method that Jean Nicolas Smogolenski and Xue Fengzuo adopted when they compiled Bi Li Shu Biao the method of five homes six into?
We checked all other logarithm values again, and found the values which first digit after decimal point are 6, 7, 8, 9 were carried, and the others values were not carried. So the previous conjecture must be true.
We calculated all ten thousand logarithm values with new carry method one more time, the results are shown as Table 3.
Comparing logarithm values in Table 3 with logarithm values in Bi Li Shu Biao, it is easy to find that there are only several different values. Exactly the different values only are 76. They are shown as Table 4.