Was 'Jade Mirror of the Four Unknowns' published in 1303, and how do we know?

Question

Wikipedia gives the date for Chinese mathematician Zhu Shijie's Jade Mirror of the Four Unknowns (四元玉鉴, Siyuan yujian) as 1303, citing

Elman, Benjamin A. (2005). On their own terms science in China, 1550-1900. Cambridge, Mass.: Harvard University Press. p. 252. ISBN 0674036476.

That reference says

On the other hand, the editors had not yet recovered Zhu Shijie's Jade Mirror of the Four Unknowns (Siyuan yujian) and his 1299 Primer of Mathematics in time for inclusion in the Qianlong Imperial Library.

Searching Google Books and Amazon Look Inside I do not find any other references in this work to the year 1303.

I'd like to know where this information comes from to be so casually dropped into the text. Usually dates like these are hard-won pieces of information, stitched together from different calendars or more obscure references, and scholars generally leave information on how they determine these things. I would expect a date to be expressed something like Dàdé 7 (the seventh year in the second era of Temür Khan's reign; note I'm neither a historian nor familiar with Chinese dates).

Answer 1

Towards the end of one of the prefaces of the book (there were two prefaces, 前序 and 松庭先生後序), it states:

大德癸卯上元日，臨川前進士莫若序。

which can be roughly translated as:

On the day of Shangyuan (lit. first full moon festival) in the Kuí-Mǎo year during Dà-Dé era, 臨川前進士 (this was most likely the honorary title of the author of the preface) presents this preface.

大德 (Dà-Dé) was the era name of emperor Chengzong of Yuan, otherwise known as Temür Öljeytü Khan. 癸卯 (Kuí-Mǎo) was the ganzhi name (see Sexagenary cycle for more information) of the year. Knowing that one sexagenary cycle accounts for 60 years, it thus becomes quite simple for us to deduce which year it was -- since 1963 CE was the most recent year in the cycle bearing the same ganzhi and we know that emperor Chengzong ruled from 1294 to 1307, therefore the preface must had been written in 1303, the only year that fell into Chengzong's reign and took the ganzhi 癸卯.

Obviously it would be another matter as to whether chapters of the book were indeed composed in the same year, but as far as we can draw from the original text, 1303 was indeed the year Zhu finalized his three volumes.

For reference, I found the original Chinese text of Jade Mirror of the Four Unknowns on Chinese Wikisource, and there's also a listing of the aformentioned preface from the digital library Chinese Text Project.