There are islands labelled as "Argonaut I.", "Dagelet I.", "Hornet In." and "Oki Shima" in the Japan Sea. Argonaut island was drawn in a broken line, which may indicate that it was a phantom island. The most noteworthy point in this map is a national border, a pink coloured broken line - it runs between Argonaut island and Dagelet island (Ulleungdo), so the map shows that Ulleungdo (Dagelet I.) and Liancourt Rocks (Hornet In.) were Japanese territory.
Of course it was a mistake - Ulleungdo was Korean territory. However, the reason why the same mistake was repeatedly adopted in western maps is quite simple. They thought Argonaut island was Take island (or Taka island) and Dagelet island was Matsu island as many western maps show. And it may have been transmitted to western mapmakers that Take island (Takeshima) was given back to Korea in the late 17th century and Matsu island (Matsushima) remained in Japan. So they may have kept publishing maps that show Argonaut island (Take island) was Korean territory while Dagelet island (Matsu island) was Japanese territory.
Most importantly, this problem is about Ulleungdo, not about Liancourt Rocks. Nobody thought that Liancourt Rocks belonged to Korea. It seems that it was a kind of a common sense in those days.
This map may be expressed in Kaneganese's method as follows:
○ 1897 British map of Japan (Popular Atlas) Three isalnds. National border between Argonaut and Dagelet.
Broughton Bay is labelled as "BROUGHTON BAY OR GULF OF KOREA".