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In present day, Istanbul spans over the two sides of the Bosphorus, and it can claim to be the largest transcontinental city in the word.
However, in classical and medieval times the city of Constantinople and its ancestor Byzantium were only in a small part of the European side of modern Istanbul. Therefore, Constantinople was in Europe.
Upvote:11
As Pere already explained (+1), Byzantium (and then Constantinople) was centered around the present-day neighbourhood of Sultanahmet on the European side of the Bosphorus, where many the oldest monuments in Istanbul can still be found (the Basilica cistern, hippodrome, the Milion, and the column of Constantine). The city then extended to the other side of the Golden Horn (Pera/BeyoΔlu, where Genoese and Venetian merchants were based) and to the Asian side.
There was a Greek settlement on the other side of the Bosphorus (Chalcedon, the present KadΔ±kΓΆy) but it was much more difficult to defend and changed hands frequently, ultimately being conquered by the Ottomans a full century before Constantinople.
Interestingly, the concept of Europe was originally used to designate exactly this region, Thrace and the western shore of the Aegean sea, long before anything resembling our modern idea of the European continent formed. So there is a case to be made that Constantinople was not only at the fringes of Europe like Istanbul is today but that it was the very definition of Europe.