2

For instance, the sentences like, "By planting more trees, we can save the earth." and "After drinking wine, he danced."

2

In the first example ("By planting more trees"), the verbal noun planting takes a direct object in the accusative, for which A&G notes we should rather use the gerundive:

The gerund with a direct object is practically limited to the genitive and the ablative (without a preposition); even in these cases the gerundive is commoner.

As noted in the same source, when choosing the gerundive, the direct object should take the same case; which in our case is the ablative case ("By planting trees..."), thus:

"serendis arboribus pluribus, possumus orbem terrarum servare"

[If we want to stick to the less-common gerund (which is only possible since we are in the ablative case): serendo arbores plures (here the direct object has its usual form in the accusative)]

It seems we can apply the same method for the second example; but if we are not confined to the gerundive, we might use a simple verb (in that case we have specific subject) ["postquam vinum bibit ...], or a participle construction (noun + participle) like "post natos homines" (after the creation of man) or "ab urbe condita", in our case "post vinum bibitum, saltavit"

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