I read that essay. I don't believe it, or rather, I think it exaggerates what it meant to study a field. I'll base my view on what "1st year calculus" means. (Do note that my high school in Miami offered matrix algebra and differential equations, so I see "1st year calculus" as being a bit basic for what a high school might offer.)

I know that in the 1800s few schools taught the calculus we now call "calculus." Back then that term was used for many fields of math, including logic ("the propositional calculus"). Nowadays it's almost only applied to differential and integral calculus. The rigorous foundations of calculus were only laid starting in the mid-1800s, so I seriously doubt that any "backwoods" student would have been doing anything remotely like ε/δ proofs then; that being the basis for our modern understanding of calculus. It might also have been that "1st year calculus" is what we now call "pre-calculus."

That's not saying it's impossible. Here's an example book on calculus from De Morgan in the early 1800s: http://books.google.com/books?id=zA4TAQAAMAAJ&ots=iIzsbl... . This is from a book series meant for adults/non-researchers interested in learning more about the latest science. You can see it has a more geometric approach. I find fascinating the footnote remarking the lack of consensus on how to use the integral sign. The comment on page 52 is telling:

There is then in this method neither the rigor of geometry, nor that approach to truth, which, in the method of Leibnitz, may be carried to any extent we please, short of absolute correctness. We would therefore recommend to the student not to regard any proposition derived from this method as true on that account; for falsehoods, as well as truths, may be derived from it.

This is quite simply because in 1836 they didn't know how to handle infinities correctly. No modern calculus book would have that phrase, because we do.

But let's suppose his father did study differential and integral calculus. That would make him the exception, and not indicative of the average. For evidence: http://pballew.net/mathbooks describes math education in the US and mentions that in 1871 for the University of Nebraska in Lincoln:

"It was early accepted that mathematics, in particular calculus, would not be a required study for all students. In the first University Catalog it is noted that "Surgery and Calculus are not required of students in the Classical course." Exemptions from other mathematics were soon included."

I also read in "The calculus for engineers and physicists" Griffin 1897 this line "...methods of cheating the student into using the Calculus without his knowing that he is doing so..." which tells me that calculus wasn't that common in 1897 either.

I could apply the same questions to what "geology" means. Here's a textbook from the latter part of the 1800s. It goes into detail about all sorts of different rocks, where they are found how they are use commercially, and so on. It describes land forms. But what of geological processes? Nothing - because we hadn't figured it out yet! It says that folded strata probably come from cooling of the earth's "heated nucleus." In the section "Drift or Boulder Period" it notes what we now say is evidence of the Ice Age, but which then was a curious unknown "produced by Icebergs, Glaciers, or both." They didn't know if it was ice covering the surface, or if the land submerged and icebergs came.

They certainly didn't know about continental drift.

So the geology of the 1800s would have been quite different from the "earth science" I learned in school.