Signs are small measurable things, but interpretations are illimitable, and in girls of sweet, ardent nature, every sign is apt to conjure up wonder, hope, belief, vast as a sky, and coloured by a diffused thimbleful of matter in the shape of knowledge. (1.3.6)
Dorothea is the "girl of sweet, ardent nature" being referred to in this passage. She's ready to "interpret" every "sign" given to her by Mr. Casaubon before their marriage in the best possible way, according to her own "wonder, hope, [and] belief."
"I should see how it was possible to lead a grand life here – now – in England." (1.3.14)
Dorothea's dreams, hopes, and plans have to do with leading a "grand life," but she's not sure how to reconcile that ambition with the realities of everyday life. She's read a lot of histories of people who led "grand li[ves]," like Saint Theresa, but she's not sure how it can be managed in the here and now.
"There would be nothing trivial about our lives. […] It would be like marrying Pascal. I should learn to see the truth by the same light as great men have seen it by." (1.3.14)
Dorothea hates "trivial" things like worrying about clothes, social visits, and what to have for dinner. She wants to spend all day thinking about "truth" and the nature of the universe. She imagines that marrying Casaubon will be like marrying "Pascal" (a great French philosopher), because he will teach her to "see the truth by the same light as great men have seen it by."
"Fad to draw plans! Do you think I only care about my fellow-creatures' houses in that childish way? I may well make mistakes. How can one ever do anything nobly Christian, living among people with such petty thoughts?" (1.4.23)
Dorothea's friends and family ridicule her passion for improving the cottages that poor people live in. She finally loses her temper – who likes having their noble and deep-seated ambitions ridiculed?
Those provinces of masculine knowledge seemed to her a standing-ground from which all truth could be seen more truly. (1.7.6)
Dorothea longs to understand the world, and because she doesn't know Greek or Latin (languages usually only taught to men at the time), she assumes that understanding "those provinces of masculine knowledge" will give her a better perspective from which to see the "truth" about the nature of the universe.
Perhaps even Hebrew might be necessary – at least the alphabet and a few roots – in order to arrive at the core of things, and judge soundly on the social duties of the Christian. (1.7.6)
This passage is just dripping with irony. Dorothea thinks that learning a smattering of Hebrew as well as Latin and Greek will help her "arrive at the core of things." Learning foreign languages might be useful and fun, but it doesn't teach you the answer to "life, the universe, and everything" (to quote Douglas Adams).
[…] there was nothing for her to do in Lowick […]; she would have preferred […] finding that her home would be in a parish which had a larger share of the world's misery, so that she might have had more active duties in it. (1.9.35)
Dorothea is so eager to help people that she's actually disappointed to find that the poor people in Lowick are actually pretty well taken care of already. She's ready to save the world, and is sad to find that it's already been saved – in this neighborhood, at least.
Such was Lydgate's plan of his future: to do good small work for Middlemarch, and great work for the world. (2.15.8)
Lydgate's dreams and plans are as lofty as Dorothea's: he wants to do good work on a local level in the town of Middlemarch, and to do great work on a grander level "for the world." He wants to save lives and cure sick people in Middlemarch and also to make a great scientific discovery that will advance the knowledge of the world.
character too is a process and an unfolding. (2.15.9)
Characters in Middlemarch develop from beginning to end – very few characters, even minor ones, remain static. Their hopes and dreams are adjusted accordingly: by the end of the novel, Dorothea learns to reconcile her ambitions with the realities of her social position.
Rosamond, in fact, was entirely occupied not exactly with Tertius Lydgate as he was in himself, but with his relation to her; and it was excusable in a girl who was accustomed to hear that all young men might, could, would be, or actually were in love with her, to believe at once that Lydgate could be no exception. (2.16.60)
Rosamond's hopes and dreams are to make everyone in love with her. She is the center of her own little world and wants to be the center of everyone else's, as well. She's more concerned, therefore, about Lydgate's "relation to her" than about his inner life or his hopes and dreams.
[A mirror] will be minutely and multitudinously scratched in all directions; but place now against it a lighted candle as a centre of illumination, and lo! the scratches will seem to arrange themselves in a fine series of concentric circles round that little sun. It is demonstrable that the scratches are going everywhere impartially, and it is only your candle which produces the flattering illusion of a concentric arrangement […] these things are a parable. (3.27.1)
Rosamond might be the center of her own little world, but, here, the narrator suggests that everyone is the center of his or her own little world. She compares the tendency that most people have to assume that everything revolves around themselves to the way the tiny scratches on a mirror appear to form perfect concentric circles around a light source. (This is true! Try it at home.)