Mr Brooke wondered, and felt that women were an inexhaustible subject of study, since even he at his age was not in a perfect state of scientific prediction about them! (1.4.49)
There were several passages describing women as a subject of scientific study by men in the "Women and Femininity" section, too. Why are women so often associated with "scientific prediction" and study? Well, the implication is that men often like to make generalizations about all women (as though women were all the same) and call it "scientific observation."
Even with a microscope directed on a water-drop we find ourselves making interpretations which turn out to be rather coarse; for whereas under a weak lens you may seem to see a creature exhibiting an active voracity into which other smaller creatures actively play as if they were so many animated tax-pennies, a stronger lens reveals to you certain tiniest hairlets which make vortices for these victims while the swallower waits passively at his receipt of custom. In this way, metaphorically speaking, a strong lens applied to Mrs Cadwallader's matchmaking will show a play of minute causes producing what may be called thought and speech vortices to bring her the sort of food she needed. (1.6.71)
This is the first of many references to microscopes, telescopes, and other lenses and scientific instruments in Middlemarch. This passage compares the interpretation of the behavior of single-cellular organisms under a microscope with the behavior of Mrs. Cadwallader, the neighborhood gossip. The narrator says that scientists are always adjusting their interpretations. Every time they get a stronger lens, they're able to make better inferences about what the single-cell organism is really up to. In the same way, you need to look closely at Mrs. Cadwallader in order to judge her correctly – she's not gossiping because she's mean, but because it's "the sort of food she needed."
But of Mr Brooke I make a further remark perhaps less warranted by precedent – namely, that if he had foreknown his speech, it might not have made any great difference. To think with pleasure of his niece's husband having a large ecclesiastical income was one thing – to make a liberal speech was another thing; and it is a narrow mind which cannot look at a subject from various points of view. (1.7.20)
Mr. Brooke is often accused of being inconsistent. He's happy about the fact that Mr. Casaubon, his niece's husband, has "a large income" from his job as a clergyman, and yet he makes a political speech criticizing the Church of England for giving too much money to clergyman. But the narrator, instead of saying, "Yep, he's inconsistent," suggests (perhaps sarcastically) that Mr. Brooke is just being a good scientist; he's trying to look at the problem from "various points of view."
"Somebody put a drop [of Mr. Casaubon's blood] under a magnifying-glass, and it was all semicolons and parentheses." (1.8.32)
Here's another lens metaphor! Mrs. Cadwallader is making fun of Mr. Casaubon to Sir James Chettam. She thinks that he's too old and passionless to marry a pretty young girl like Dorothea, and Sir James agrees. And since the common belief in the early nineteenth century was that passion came from the blood (which is where we get the expression "hot-blooded" to mean "passionate"), Mrs. Cadwallader illustrates her point by suggesting that Mr. Casaubon doesn't have any blood at all – it's all punctuation marks because that's all he cares about.
He was ambitious of a wider effect: he was fired with the possibility that he might work out the proof of an anatomical conception and make a link in the chain of discovery. (2.15.6)
Mr. Lydgate's scientific ambitions extend beyond the Fever Hospital or his Middlemarch patients: he wants to discover something new about the basic unit of life that will make a "link in the chain of discovery." He wants to leave something behind him that will benefit the scientific community forever.
There was fascination in the hope that the two purposes would illuminate each other: the careful observation and inference which was his daily work, the use of the lens to further his judgment in special cases, would further his thought as an instrument of larger inquiry. (2.15.7)
Here's a reference to a "lens" that is literal, not metaphorical: Lydgate wants his scientific research (using the "lens" of his microscope) to parallel his "daily work" as a medical practitioner. He hopes that the general work he'll do as a researcher will benefit the particular cases he'll work on as a doctor, and vice versa. Eliot is suggesting that it can be beneficial to move back and forth between the particular and the general – which is exactly what she often does as a narrator.
No man, one sees, can understand and estimate the entire structure or its parts – what are its frailties and what its repairs, without knowing the nature of the materials. (2.15.8)
Part of Lydgate's ambition is to discover the basic tissue of all life. He believes that in order to understand the "structure," you first have to know "the nature of the materials" that make it up. This could perhaps be paralleled to the way Eliot shows us the "nature" or personality of her characters in order to clarify their actions. It's impossible to understand a person's actions without first having a glimpse of their inner life.
And he counted on quiet intervals to be watchfully seized, for taking up the threads of investigation – on many hints to be won from diligent application, not only of the scalpel, but of the microscope, which research had begun to use again with new enthusiasm of reliance. (2.15.8)
In order to carry out his ambition of scientific discovery, Lydgate will use the "microscope" as well as the "scalpel." For a long time, scientists relied more on dissection of samples than on examining them under microscopes. Lydgate plans on using both techniques – he'll look at things up close, on a micro level, as well as from a macro level. Again, Eliot suggests that it's beneficial to combine the micro and the macro, or the particular with the general, in order to understand something thoroughly.
henceforth he would take a strictly scientific view of women, entertaining no expectations, but such as were justified beforehand. (2.15.25)
Here's yet another passage suggesting that women can be understood scientifically. The assumption, of course, is that you can make generalizations about all women. Lydgate's tendency to apply scientific observation and objectivity to his personal relationships is part of why his relationship with Rosamond crumbles.
"I was early bitten with an interest in structure." (2.17.26)
Lydgate is interested in the way things are put together. He's more interested in how and why things work than in what they do. The same is true of his view of people. He's more interested in looking at why you're feeling the way you are than in offering a sympathetic shoulder to cry on.
[…] his home preoccupation with scientific subjects, which seemed to her almost like a morbid vampire's taste. (7.64.90)
There are relatively few passages in the novel that describe science from the point of view of a non-scientist. This is one of those few. Rosamond thinks that her husband's obsession with anatomy is "morbid" and "vampir[ic]." This wasn't uncommon at the time – in the 1830s, there was a lot of anxiety about the passage of a law called the Anatomy Act, which allowed certain human bodies to be dissected. Some Middlemarchers are afraid that Lydgate is just waiting for them to die so that he can dissect their bodies. That desire to "feed" off of human bodies (metaphorically speaking, of course – he's no cannibal) might be related to Rosamond's opinion that Lydgate's scientific pursuits reflect a "morbid vampire's taste."