Based on accounts of various ancient writers, scholars have painted a sketchy picture of the activities of an all-female cult that, perhaps as early as the sixth century B.C., worshipped a goddess known in Latin as Bona Dea, “the good goddess.”
(A) Based on accounts of various ancient writers (Scholars can't be based on various ancient writers , it is the action "have painted" which is based on accounts)
(B) Basing it on various ancient writers’ accounts ( Basing IT on .............. here IT(pronoun) refers to the action "have painted" which is wrong to the core as pronoun can ONLY REFER to NOUN or another PRONOUN
(C) With accounts of various ancient writers used for a basisScholars DID something with accounts of various ancient writers --- It changes the meaning and it is wordy too.
(D) By the accounts of various ancient writers they usedScholars didn't use the accounts of various ancient writers, they based their action on accounts of ancient writers.
(E) Using accounts of various ancient writersPerfect as -ING modifier makes sense with the scholars as
Scholars DID something USING blah blah blah......
Uses of -ING.
(1). To modify a NOUN/PRONOUN
(2). To Modify VERB and VERB's subject
(3). Result of the main clause where main clause is considered as NOUN PHRASE
I Assisted You => KUDOS Please
Done with the wedding, extremely happy