However, as this map is a large one, it would be good to learn the locations of these islands.
The three longitudinal lines in the close-up map (above) indicate the longitude 129 degree, 131 degree and 133 degree (E) respectively from west to east, so you will find Taka Island, that is Argonaut island, locates at less than 130 dgree (about 129 degree 50'). Also you'll find Matsu Island, Dagelet Island or the true Ulleungdo, is located at less than 131 degree (about 130 degree 50'), Liancourt Rocks at less than 132 degree (about 131 degree 50').
Here is a Japanese map called "新撰朝鮮国全図" (Newly Edited Whole Map of Chosun Country) which was published in 1894. It has been used by pro-Korean people to insist that Japan admitted Dokdo to be Korean territory. There are two islands in the Sea of Japan - 竹島(Takeshima) and 松島(Matsushima), which pro-Korean people insist as Ulleungdo and Liancourt Rocks (Takeshima/Dokdo).
If the map below is too small to examine, please visit here: http://www.geocities.jp/tanaka_kunitaka/takeshima/senshin-1894/If you examine the map precisely, you will notice that the longitudinal line beside Takeshima runs beside Iki island at south, which locates at around 129 degree 40' - so this longitudinal line may mean longitude 130 degree. Then it may mean "Takeshima" in the map is located at about 130 degree (129 degree 50') - it coincides with Argonaut island which was mentioned above. The next longitudinal line runs through Kitakyushu (Kokura) that locates at 131 degree. "Matsushima" in the map is located at the middle of the two lines - it seems as it locates at about 130 degree 40', which is almost the place of Ulleungdo although not very accurate. By the way, as Liancourt Rocks locate at about 131 degree 50' as we showed above, so Liancourt Rocks is out of this map.
But there is a problem - the map was not drawn in east longitude based on the Greenwich point but it was drawn in west longitude based on the Japanese standard point at Azabu, Tokyo (139 degree 44' 28" E). The island "Takeshima" in the map is located beside the line "九". The Chinese number "九" means "9 degree W" here, which is similar to "130 degree 44' E". But if this line means "130 degree 44' E", it must run through Kitakyushu because Kitakyushu is located at 130 degree 50', but the line crossed Saga prefecture, one degree west. Instead, the line "八" (8 degree W) which is 131 degree 44' E crossed Kitakyushu. So you can understand that this map is incorrect, as whole the Japanese figure in the map was drawn mistakenly just one degree east. Then, the problem is that we can't decide whether the two islands "Takeshima" and "Matsushima" were also drawn one degree east or not.
1) If the two islands were also drawn one degree east, "Takeshima" in the map (slightly less than 130 degree 44' E, maybe around 130 degree 40' E) should be at about 129 degree 40' E. This is almost the same place as Argonaut island (129 degree 50' E). "Matsushima" which is located at the middle of the lines 8 degree W and 9 degree W (more precisely slightly eastwardly shifted from the middle point; almost 131 degree 30' E) should be 130 degree 30' E which is close to Dagelet island (130 degree 50' E).
2) If the two islands were drawn just the exact places, "Takeshima" is located at about 130 degree 40' E, which is near to Dagelet island (130 degree 50' E) while "Matsushima" is located at 131 degree 30' E, which is almost the middle point between Dagelet island and Liancourt Rocks.
By the way, the east limit of the map is 西経八度 (= 131 degree 44' E) which excluded Liancourt Rocks (131 degree 50' E) - if they tried to draw Liancourt Rocks, the rocks were to be drawn in the margin of the map. But there is no such rocks were drawn in the map.
Considering other maps with Takeshima and Matsushima from the same era (see the references), 1) should be most likely the case. Even if 2) was the case, it didn't always refer to Liancourt Rocks.
東京地学協会編 朝鮮全図 (1894; http://www.tanaka-kunitaka.net/takeshima/tokyo-1894/ ) with Ulleungdo - Matsushima at slightly west than 131 degree E,
朝鮮全岸 (1896; http://www.geocities.jp/tanaka_kunitaka/takeshima/koreacoast-1896/ ) with Ulleungdo- Matsushima at around 131 degree E,
日本全図 (1877; http://www.geocities.jp/tanaka_kunitaka/takeshima/nihonzenzu-1877/ ) with
Takeshima at slightly west of 西十度 line = 129 degree 44' E and Matsushima at slightly west of 西九度 line = 130 degree 44' E (which means these are Argonaut and Dagelet islands)
As a result, "竹島(Takeshima)" in the map is highly likely Argonaut island, whcih western maps sometimes called as "Take Isl" or "Taka Isl." while "松島 (Matsushima)" in the map is highly likely true Ulleungdo or Dagelet island, which western maps sometimes called as "Matsu Isl." - so these two islands in the map coincided with "Taka Isl" and "Matsu Isl" in the American map.
Of course, no one can say from this map that Japan admitted Dokdo to be Korean territory.