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Here is a short post about it in English (http://maviboncuk.blogspot.com/2015/07/nacib-k-djezvedijan.html) .
I see:
Damasios / Phylo??polous / le 4 Nov 1922 / Constantinople
The top two lines are definitely a Greek personal name, though I may not have it quite right. At that time, ethnic Greeks were a plurality of the population in Constantinople.
It was a troubled time: Constantinople was still under occupation by the British, French, and Greek armies, but Ataturk's Republican armies had just destroyed the Greek armies in Asia Minor - this watch was engraved just a few weeks after the armistice in late October 1922. It is easy to imagine its Greek buyer having having some trepidation about the political situation, which was not unreasonable considering the huge forced migrations of Greeks and Turks after the treaty of Lausanne in July 1923. An interesting time in an interesting place.
I hope I won't bore you all with more historical sociology, but it is also interesting to think about the use of French by an ethnic Greek in such a personal inscription at this place and time. Surely he spoke Greek, probably Turkish as well. But French was (literally) the lingua franca of educated Turks, Jews, Armenians, Greeks, and foreigners in the Ottoman period, and anyone who was educated would learn it (not unlike the rest of Europe at the time). It's plausible that the whole watch transaction took place in French, though then as now the merchants of the Grand Bazaar would no doubt speak to you in pretty much any language you like.
France had a long relationship with the Ottoman state going back to the 1550s, and cultivated a political relationship that had both an economic dimension (playing a large role in financing the Ottoman debt) and a religious dimension (in France's desire to play the protector of Christians in Jerusalem and Ottoman Christians more generally). Inasmuch as the Ottoman elite were looking to Europe in the 19th century, they looked to France for education. For instance Osman Hamdi Bey, the great Ottoman painter, archaeologist, and administrator, spent 9 years in Paris first as a law student, then as an apprentice to the painters Gerome and Boulanger. In a sense Paris was one of the most important sites of Ottoman modernisation.
Anyhow, this is all to point out the extent to which educated Ottomans of any ethnicity internalized the French language as a marker of class and identity. So that when Mr. Phylo??polous bought this lovely watch from Dzevedjian, he chose to memorialize his watch purchase in that language, rather than Greek. In a personal inscription that hardly anyone but him would see, we have to imagine that using French was meaningful to him as a marker of his identity as an educated person of the metropolis.
Ok, now back to your regularly scheduled watch programming!
The name appears to be
Anastasios Phylactojsoulos. Not sure about the last name.
Anastasius (Latinized) or Anastasios (Greek: Αναστάσιος, romanized: Anastasios) is a masculine given name of Greek origin derived from the Greek word ἀνάστασις (anastasis) meaning "resurrection".
A phylactery, seems to be associated in ancient Greek to protection of the souls. Maybe something like an amulet. It is also found in Hebrew where the person in mourning wears a small box like item strapped to their forehead.