imskpwr wrote:Among the objects found in the excavated temple were small terra-cotta effigies left by supplicants who were either asking the goddess Bona Dea's aid in healing physical and mental ills or thanking her for such help.

(A) in healing physical and mental ills or thanking her for such help
(E) to heal physical and mental ills or thanking her for such help

Is there any meaning change?

AID IN VERBing = AID in the PROCESS of the VERBing action.
Supplicants who were asking the goddess Bona Dea's AID IN HEALING physical and mental ills.
Here, what is being requested is Bona Dea's AID in the process of HEALING.

AID TO X = aid is being GIVEN TO X.
The United States provides aid TO MANY EMERGING NATIONS means that AID is being given TO THE NATIONS.
In the incorrect answer above:
Supplicants who were asking the goddess Bona Dea's AID TO HEAL physical and mental ills.
This makes no sense: aid cannot be given TO A VERB (heal).

Also:
to ask + DIRECT OBJECT + INFINITIVE.
This construction is appropriate when the DIRECT OBJECT is being asked to perform the infinitive action:
Mary ASKED JOHN to wash the dishes.
Here, JOHN is begin asked TO WASH the dishes.
In the incorrect answer above:
Supplicants who were ASKING THE GODDESS BONA DEA'S AID to heal physical and mental ills.
Here, the implication is that the AID ITSELF is being asked to heal the ills.
Not the intended meaning: the AID itself cannot heal.

imskpwr wrote:Among the objects found in the excavated temple were small terra-cotta effigies left by supplicants who were either asking the goddess Bona Dea's aid in healing physical and mental ills or thanking her for such help.

(A) in healing physical and mental ills or thanking her for such help
(E) to heal physical and mental ills or thanking her for such help

Is there any meaning change?

Yes.

With A:
supplicants were asking the goddess Bona Dea's aid in healing physical and mental ills.

This implies that supplicants were asking the goddess Bona Dea's aid so that physical and mental ills can heal.

With E:
supplicants were asking the goddess Bona Dea's aid to heal physical and mental ills .

Let us take an easier example: The poor asked for Financial help to become self-reliant.

This means that the poor asked for Financial help so that the poor can become self-reliant.

Along similar lines, option E can imply that supplicants were asking the goddess Bona Dea's aid so that the "supplicants" can heal physical and mental ills. This changes the meaning, because the original sentence does not talk about "supplicants" doing the healing.