The 1840 map of Japan to the left was made by German physician and traveler Philipp Franz Balthasar von Siebold. During his travels Dr. Siebold spent six years (1823 - 1829) in Japan, where he acquired some detailed maps of Japan and Korea made by the Japanese surveyor and cartographer Ino Tadataka. Siebold combined his knowledge of both Western and Japanese maps to make the map to the left, which is entitled "Karte vom Japanischen Reiche, nach Original-karten und astronomischen Beobachtungen der Japaner : die Inseln Kiu Siu, Sikok und Nippon."
One big problem with the Siebold map is that it included the non-existent island of Argonaut, which was a mismapping of the Korean island of Ulleungdo by the captain of the British ship "HMS Argonaut" in 1789. British Captain James Colnett mismapped Ulleungdo at the coordinates 37º 32' N and 129º 50' E, resulting in the non-existent island of "Argonaut" showing up on many Western maps about 110 kilometers to the northwest of the real Ulleungdo, which at the time was referred to by many Western nations as "Dagelet Island."
Dr. Siebold then labeled the non-existent Argonaut as "Takeshima," which was the Japanese name for Ulleungdo, and the real Ulleungdo as "Matsushima," which was the Japanese name for Liancourt Rocks. Dr. Siebold seems to have mistakenly shifted the names he saw on Japanese maps up from Ulleungdo and Liancourt Rocks to Argonaut and Ulleungdo, respectively.
Dr. Siebold's mistake caused a lot of confusion for some Japanese, who started to assume the non-existent island of Argonaut was Korea's Ulleungdo and the real Ulleungdo was Japan's Matsushima (Liancourt Rocks). The confusion came to a head in 1876 when Japanese businessman Mutoh Heigaku (武藤平学) petitioned the Japanese government HERE to develop Matsushima, which he believed to be Japanese territory.
As part of discussions in the Japanese Ministry of Foreign Affairs to determine the location of Mutoh's Matsushima, Bureau of Documents Director Watanabe Kouki (渡辺洪基) provided two opinions, HERE and HERE, on the possible location of the island.
Below is a close-up of the portion of the map that shows Ulleungdo (Matsushima) and the non-existent island of Argonaut (Takeshima).