The terms "flat character" and "round character" were first defined in E.M. Forester's Aspects of the Novel. A flat character is a minor character who does not change during the course of the story. A round character is a major character with a complex personality who develops throughout the story. A round character is a slightly different person at the end of the story than he or she was at the beginning. The degree to which a character is flat or round is often a matter of debate.
When Captain Brown appears at the beginning of "Our Society at Cranford," Miss Jenkyns reacts negatively to his intrusion into what is primarily an all-female society. However, her opinion changes when Brown displays his pleasant, well-mannered personality. He then demonstrates his heroism when he pulls a child from the path of a train and is killed in the process.
One could argue that Brown is a flat character, as he is only present for a small part of the story, during which his character remains fairly constant. However, he reveals different aspects of his personality at different moments and undergoes the most profound change a character can: death. While his presence in the story affects the other characters, his absence has a profound impact on the lives of his two daughters and Miss Jenkins, who becomes their guardian. Thus he is a catalyst for the changes that occur in Gaskell's round characters.
There is at times disagreement over what characters fit the definitions of flat and round characters. Understanding the definitions is helpful.
Aside from flat stock characters (the foil) and stereotypical characters (mean stepmother), flat characters who are poorly or briefly developed have rigid physical descriptions (with green eyes that flashed golden (always?)); have a narrow range of emotions, feelings, thoughts; have simple motives; lack psychological (the interplay of cognition and emotion) depth and complexity. In short, flat characters are developed within narrow and restricted patterns and, in addition, may be identified by a particular single quality, idea, or trait (shifty eyes).
Round characters, on the other hand, are developed with, envisioned with, and endowed with a flexible physical description; a full range of emotions, thoughts, feelings, ideas, motives; they have complex psychologies with depth and complexity. In addition, they are not identifiable by one particular single quality, idea, or trait.
Captain Brown may threaten to be a flat character at his first introduction since his main function then is to declaim his poverty in the streets. However as his involvement in the ladies' lives and in the story deepens, he is shown to have a full range of characteristics, as enumerated above. An example is the range of emotions and reactions he experiences around the combative readings of Mr. Boz and Dr. Johnson.
Further, a deep psychological struggle is introduced as he distracts his attentions from his friends at church until after he has attended to his elder daughter. Therefore, Captain Brown is a round character, but a subsequent discussion may address the author's skill in portraying her characters, even though they are round.