I don't have much to add to the above replies other than, as a rule of thumb, participles
nearly always take the case of their subjects. Some sentences can be so complex that finding
their subjects, be they implicit or explicit, isn't always an easy task.
Here's such an example from Plato. Apology 21e-22a:
μετὰ ταῦτ᾽ οὖν ἤδη ἐφεξῆς ᾖα, αἰσθανόμενος μὲν [καὶ] λυπούμενος καὶ δεδιὼς ὅτι ἀπηχθανόμην,
ὅμως δὲ ἀναγκαῖον ἐδόκει εἶναι τὸ τοῦ θεοῦ περὶ πλείστου ποιεῖσθαι—ἰτέον οὖν, σκοποῦντι τὸν
χρησμὸν τί λέγει, ἐπὶ ἅπαντας τούς τι δοκοῦντας εἰδέναι.
Socrates is on a journey to find whether he indeed the wisest of them all as revealed by the oracle.
After these events, I now made my way from one to another, realizing, grieving, and fearing that
[in doing so] I incur their hatred, nevertheless feeling obligated to do god's work above all --
it was incumbent on me then to go to all those who were reputed to know something and examine the
The subject of the first three participles is in nominative, agreeing with the subject of ᾖα, ἐγώ =
Socrates. The parenthetical sentence though has a participle in dative, because it agrees
with the implicit subject μοι of the verbal adjective ἰτέον. In literary Hebrew we have a similar
usage, שומה עלי, which roughly translates to "it is incumbent on me."