Participles of Means and Cause

Participles can function as either an adjective, adverb, or noun (depending on context [and technically a gerund in English speak]), which allows for various endings (e.g., -ing, -ed, -en,-d, -n, -t). 

This post focuses on two functions of participles that modify verbs. Therefore, they will be used as adverbs. Rather than being boring and translating with a simple -ing, it is more helpful to insert a word or two to help the reader see the connection between the dependent and independent clause (assuming that connection is clear). If it's not, then leaving it plain vanilla is probably best.

Means: Extremely common use of participles, and answers "How an action is accomplished." This is distinct from manner, which gives an accompanying characteristic or attribute of the person performing the action (e.g., "rejoicing").

Perhaps my most favorite example is 1 Peter 5:6-7.

Ταπεινώθητε οὖν ὑπὸ τὴν κραταιὰν χεῖρα τοῦ θεοῦ, ἵνα ὑμᾶς ὑψώσῃ ἐν καιρῷ, 7  πᾶσαν τὴν μέριμναν ὑμῶν ἐπιρίψαντες ἐπ᾽ αὐτόν, ὅτι αὐτῷ μέλει περὶ ὑμῶν.  "Humble yourselves, therefore, under the mighty hand of God so that at the proper time he may exalt you, "by means of/by" casting all your anxieties on him, because he cares for you."

The command is to "humble yourselves." The participial clause "casting anxiety" answers how we are to be humble. Prayer, in this case, is an act of humility. It's an expression of dependence on God. For Peter's letter as a whole, and his audience in their situation, this would be a great comfort.

ἀλλὰ ἑαυτὸν ἐκένωσεν μορφὴν δούλου λαβών, ἐν ὁμοιώματι ἀνθρώπων γενόμενος· but he emptied himself "by means of/by" taking the form of a slave, becoming in the likeness of men (Phi 2:7) The main verb is so vague, and the participle answers how Jesus emptied himself. There are other options here (result, temporal, but those are not nearly as probable--sometimes for grammatical reasons). The "problem" with means here is that "emptying" usually doesn't include "taking" or adding something to the equation. When we understand the background of Philippians 2 as an early hymn (maybe one of the earliest creeds), we can allow for poetic license.

Cause: Another extremely common adverbial use of the participle is to give the reason an action occurred. By "reason," I mean the action looking backward that caused a current reality. All tenses are adequately represented as causal participles, and Wallace notes that this participle usually precedes its main verb.

Ἰωσὴφ δὲ ὁ ἀνὴρ αὐτῆς, δίκαιος ὢν καὶ μὴ θέλων αὐτὴν δειγματίσαι, ἐβουλήθη λάθρᾳ ἀπολῦσαι αὐτήν. "But Joseph, her husband, "because he was" just/righteous and not wanting to shame her, desired to put her away privately (Mat 1:19)

For deeper explanation, see Wallace, Greek Grammar, 628-32.