what's the etymology of OBstetrics?

Well, obstetrix is a midwife, and there's a Late Latin backformed verb obstetrico 'to perform the office of a midwife, to assist in childbirth'. From ob- + sto 'to stand by'; cf. English bestead. Obstetrics and obstat are therefor cognate with one another. Obsto means 'to stand before or against (something), withstand, thwart, hinder, oppose'. A good example of how a form of the verb came to have almost an opposite meaning from hinder to help.