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This has puzzled me for some time. I've found nothing online. Here's my question …
Why do we say arithmetic, algebra, geometry, and trigonometry, but say the calculus?
I say "we," but I don't use that construction. And I encounter it more in written form than spoken. Especially in mags like SciAm, Discover, etc. That usage seems to suggest calculus deserves more respect or recognition. Puzzling. Calculus is just another branch of math. Any insights?
Just a guess here, but perhaps it's because we know who invented the technique of calculus: Sir Isaac Newton (unless you're German, in which case it was Gottfried Leibniz). Whoever gets credit as the inventor, it was presented to the world as a single unified technique for solving certain problems, and as a single technique it gets the definite article by convention.
(By the way, of the other terms you compared it with, algebra also has a definite article, and I defy any modern-day mathematician to omit it without attracting attention.)
I coulda' sworn I read about the "inventor of algebra" somewhere, so Ron's comment sent me out for a little research. Sure enough, although a few specific names are associated with forms of algebra, it appears there was no single inventor. The development of algebra was smeared out over a couple centuries and at least that many cultures. So calculus is indeed unique in that sense (Newton: integral calculus, and Leibniz: differential calculus). Might be Ron is on to something there. Like, we talk about "the transistor" and "the laser" or other physical inventions. But I still don't see the logic behind that etymology.
BTW, I've rarely heard the definite article used with algebra though, as Ron claims is the case. But then, I don't read many math papers.
Bob's comment sent me to Google Ngrams, where I found this curious result for "the calculus." It seems to be a more recent construction, peaking in the 1960s, which might have had something to do with the high visibility of math & science during the Space Race era. Unfortunately, there's no way to filter results for "calculus" alone and not pull in that same data. Of course, while I was there I looked at "the calculus of integrals" and got nothing. But "integral calculus" showed about the same pattern of usage as "the calculus."
So I go back to my contention that it's some type of affectation related to the fact that those who can do calculus feel somewhat superior to those who can't, and use the definite article to set it apart from "blue-collar math." Not like I couldn't be swayed by another explanation.
Heimhenge said:
BTW, I've rarely heard the definite article used with algebra though, as Ron claims is the case. But then, I don't read many math papers.
You misunderstood my reference. The word algebra comes from the Arabic word al-jebr meaning "reunion of broken parts" or "restoration"; the Arabic definite article al is an (ahem!) integral part of the term in English, and you never hear anyone refer to "doing my gebra homework". The term itself comes from the title of an eighth-century Arabic treatise on calculation by the author Muhammad ibn MÅ«sÄ al-KhwÄrizmÄ«, whose own name also gives us the word "algorithm".
I think it's just alternate definitions or uses of the word calculus.
Within the field of mathematics, calculus is simply referred to as "calculus", not "the calculus". e.g., "For this course, you will need to know basic calculus" or "This type of problem requires advanced calculus". I never see, "I had to use the calculus to solve this."
I only see "the calculus" used in a literary sense to mean "the math behind" or "the process of calculating." Very few, if any, of those references actually refer to the field of mathematics called calculus. Google "the calculus of" for an entertaining list of uses.
I also found that in the medical field, calculus means a specific type of tartar or plaque. Perhaps I should start reading "The calculus of happiness" as the "the plaque build-up of happiness." 
Martha Barnette
Grant Barrett
Grant Barrett
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