Herbert Weir Smyth, A Greek Grammar for Colleges

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[*] 1497. The dative participle, without a noun or pronoun, is frequently used in the singular or plural to denote indefinitely the person judging or observing. This construction is most common with participles of verbs of coming or going and with participles of verbs of considering.

““ἡ Θρᾴκη ἐστὶν ἐπὶ δεξιὰ εἰς τὸν Πόντον εἰσπλέοντι” Thrace is on the right as you sail into the Pontus” X. A. 6.4.1, ἔλεγον ὅτι ἡ ὁδὸς διαβάντι τὸν ποταμὸν ἐπὶ Λυ_δία_ν φέροι they said that, when you had crossed the river, the road led to Lydia 3. 5. 15, οὐκ οὖν ἄτοπον διαλογιζομένοις τὰ_ς δωρεὰ_ς νυ_νὶ πλείους εἶναι; is it not strange, when we reflect, that gifts are more frequent now? Aes. 3.179, ““τὸ μὲν ἔξωθεν ἁπτομένῳ σῶμα οὐκ ἄγα_ν θερμὸν ἦν” if you touched the surface the body was not very hot” T. 2.49, ““πρὸς ὠφέλειαν σκοπουμένῳ ὁ ἐπαινέτης τοῦ δικαίου ἀληθεύει” if you look at the matter from the point of view of advantage, the panegyrist of justice speaks the truth” P. R. 589c. So (ὡς) συνελόντι εἰπεῖν (X. A. 3.1.38) to speak briefly (lit. for one having brought the matter into small compass), συνελόντι D. 4.7.

a. The participle of verbs of coming or going is commonly used in statements of geographical situation.

b. The present participle is more common than the aorist in the case of all verbs belonging under 1497.