The dating of the work is fairly well pinned down. We have comments by Zhang Qiujian which criticise the accuracy of one of the solutions given in the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual). This gives 468 as the latest possible date for the work to be written. On the other hand a change of volume standard which took place in 425 in mentioned in the text so it must have been written after that date. The dates we give cannot therefore be more than 40 years in error for the author of the work.
The treatise contains three chapters in the usual style of problems and solutions. Chapter 1 contains 19 problems, chapter 2 contains 29 problems and the final chapter contains 44 problems. One significant idea which appears in the text concerns representation of numbers in the decimal notation. Xiahou Yang notes that to multiply a number by 10, 100, 1000, or 10000 all that needs to be done is that the rods on the counting board are moved forwards by 1, 2, 3, or 4 decimal places. Similarly to divide by 10, 100, 1000, or 10000 the rods are moved backwards by 1, 2, 3, or 4 decimal places. What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10.
Although Xiahou Yang has no symbol for 0 in an empty place, there is good evidence from his description of moving numbers to the right and left that he at least has a virtual zero in mind despite the lack of a symbol.
Certainly when Li Chunfeng came to edit this text to make it a suitable text for the Imperial examinations, which it became in 656, he had to correct some of the problems. In particular some of the problems dealing with areas in the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) use incorrect formulae.
Article by: J J O'Connor and E F Robertson