Learn Italian: Avendo finito il calcolo, lo scienziato ha spiegato la teoria. - Having finished the calculation, the scientist explained the theory.

The past gerund is a more advanced, elegant way to show that one action was completed before another. It efficiently expresses chronological anteriority (he finished the calculation first) and often implies a cause (because he finished it, he was ready to explain). Remember an important rule: the subject of the gerund must be the exact same as the main verb, which in this case is lo scienziato.

How do I know whether to use avendo or essendo to form the past gerund?

You use the exact same auxiliary verb that you would use for the present perfect (passato prossimo). Because finire (to finish) has a direct object here (il calcolo), it takes avere. Therefore, its past gerund uses avendo. If a verb required essere (such as a verb of motion), you would use essendo.

Does the past participle finito need to agree with the subject lo scienziato?

No, it stays as finito. Just like in other compound tenses, when the auxiliary is avere, the past participle does not agree with the subject. It would only change its ending if there were a direct object pronoun (like lo or la) attached directly to the gerund, which isn't the case here.

As a quick refresher, why is the article lo used for scienziato?