It’s circumstantial if it’s in the predicate position. That is, it doesn’t have a definite article directly before it.
For example, τὰ παρόντα is not circumstantial (it’s attributive) because it has the article τὰ directly before it.
I’m going to try the algorithm out on exercise 5. Let me know if this is helpful or makes things even more confusing. I’m trying to develop a method of handling participles as I find them confusing.
οἱ ἄφρονες τῶν ἀνθρώπων τὰ παρόντα ἀφέντες τὰ ἀπόντα διώκουσιν.
Step 1. Do we have any circumstantial participles?
Yes, ἀφέντες is not preceded directly by an article (it is in the predicate position).
Step 2. Identify the circumstantial particle (see page 220 of Mastronarde)
nom, plural, masc, aorist, active, participle, ἀφίημι.
Step 3. What is the participle’s noun? (must be nom, plural, masc)
in this step we are considering the participle as an adjective which must be in concord with a noun
οἱ ἄφρονες τῶν ἀνθρώπων is nom, plural, masc
Note: the participle’s noun is also the subject of the participle’s verbal action.
So the participial phrase is
οἱ ἄφρονες τῶν ἀνθρώπων τὰ παρόντα ἀφέντες
the fools among mankind having let go of what is at hand (non-finite verb in participial phrase)
Note: this is not a clause, as it has no finite verb
Step 4: What is the main verb clause?
οἱ ἄφρονες τῶν ἀνθρώπων τὰ ἀπόντα διώκουσιν
the fools among mankind pursue what is absent (finite verb in main clause)
Step 5: What is the relationship between the two verbal actions?
Two verbal actions one having happened before the other. This is a temporal circumstantial participle.
There is nothing concessive, causal or conditional involved.
So the English translation in this case can be two clauses joined by “and”.
the fools among mankind let go of what is at hand and pursue what is absent.
Or, we could preserve the participial nature with something like:
the fools among mankind, having let go of what is at hand, pursue what is absent.