Concerning her early work in number theory, J. H. Sampson states: "She was clever with formal algebraic manipulations; but there is little evidence that she really understood the , and her work of that period that has come down to us seems to touch only on rather superficial matters." Gray adds that "The inclination of sympathetic mathematicians to praise her work rather than to provide substantive criticism from which she might learn was crippling to her mathematical development." Yet Marilyn Bailey Ogilvie recognizes that "Sophie Germain's creativity manifested itself in pure and applied mathematics ... provided imaginative and provocative solutions to several important problems", and, as Petrovich proposes, it may have been her very lack of training that gave her unique insights and approaches. Louis Bucciarelli and Nancy Dworsky, Germain's biographers, summarize as follows: "All the evidence argues that Sophie Germain had a mathematical brilliance that never reached fruition due to a lack of rigorous training available only to men."