This informal paper, an expanded version of the short talk given by the second author at the conference on the occasion of the 300th anniversary of Seki's death, is slightly non-standard in nature and perhaps requires a short explanatory preamble. The authors are not professional historians of mathematics, and no attempt has been made to interpret the material discussed from a historical viewpoint. Instead, the first section contains several specific mathematical comments, from the point of

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... of a contemporary professional mathematician (D.Z.), on a few of the problems and solutions of Seki and of his predecessors in China and Japan, pointing out places where the mathematical content is unexpectedly naive or unexpectedly sophisticated, or where particular mathematical features of the problems permit deductions about their authors' methods or views. The second section concerns the thorny question of possible contacts that Seki or his disciple Takebe Katahiro may have had with European mathematics as a result of the Dutch presence in Dejima and their yearly visit to the Edo court. In particular, we describe the results of a search (by S.W.-Z.) through the archives of the Dutch East India Company for the years in question that yielded details of a meeting between Takebe and the Dutch but show clearly that there was, at least on this occasion, no serious discussion of any scientific or mathematical questions. We also mention a few other arguments militating against the thesis that there was any direct impact of European mathematics (prior to the partial lifting of the ban on Western books in 1720) on the work of these two scientists.