# Why is it called "the" calculus?

I have often seen people refer to this branch of mathematics as “the” calculus, but why the definite article? I’ve never seen anyone refer to “the” algebra. What’s the deal?

-b

I believe it is short for the calculus of infinitesimals. The “of infinitesimals” was dropped when the concept of limit was introduced and infinitesimals removed. At the time it was beleived that infinitesimals were not consistent. They were refered to as ghosts of departed quantities. Since that time non-standard analysis has shown that infinitesimals can be used and that they introduce no contradictions.

Interesting. The Calculus course I took in college (which was longer ago than I care to admit) was taught using infinitesimals, not limits. (It was a special section, the standard curriculum used limits.) In fact, limits were not even introduced until about halfway through the course.

Our instructor told us that when Newton developed Calculus he did so using the concept of infinitesimals but that later limits were used as being more easily understood.

I work with a good number of mathematicians (we do development for a statistical forecasting system) and many of them were amazed when they learned about my course; they had heard of the infinitesimal approach but had never seen it. Unfortunately, I lost my old textbooks in my last move so can’t confuse people by showing it to them anymore.

Well, I disagree with your instructor. Constructing infinitesimals from the reals is tricky (you have to use an ultrafilter), but constructing the reals from the integers is tricky also. You can just define them. The concept of limit is IMHO the major stumbling block of most begining calculus students.

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AFAIK, the inimitable DrMatrix is on the money with the actual answer to the OP. My trivia moment is that I was at an exhibit of the Grolier Club (a super-snobby book collector’s club in NY that has an exhibit room which we lesser beings can see for free) and they had this exhibit from this book collection from this Swiss fellow who did his collecting in the '20’s through the '60’s. It seems in about 1928, the University of Leiden (I think) was de-accessioning some books to raise cash and they sold one of their two copies of the first edition of Newton’s Principia. They sold this one because it had been marked up in the margins. It turns out that it was Leibniz’ personal copy with his notes in the margin (!!!). Given the fact that Newton is reputed to have invented (the) calculus first but kept it to himself, and only published when his colleagues pressured him into doing so in order to make sure he got the credit over Leibniz, who was publishing related materials and claiming credit, one can only imagine what is in that marginalia (I don’t read German and could not flip through the book; not surprisingly, it was behind glass) - what the heck were the folks in Leiden thinking!?!

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The al in algebra happens to be a definite article in Arabic (just as in alcohol and alchemy).
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Newton didn’t develop ‘calculus’, did he? I thought he was the one who gave it the other name (‘fluctions’?).

You rang?

Ah. If only I spoke Arabic. So what does the “gebra” translate as?

-b

Well, our class just defined them. We started by defining the “HyperReal” number system which was “All Real Numbers plus Infinitesimals”. An Infinitesimal was defined as something like “the smallest possible difference between two Real Numbers R[sub]n[/sub] and R[sub]n+1[/sub]”. We also used 1/infinity as a definition.

As for as limits being harder to understand, I wouldn’t know since I learned infinitesimals first. We used limits as another definition of an infinitesimal. For example, the limit (1/x)[sub]x->infinity[/sub] is zero. The value approaches zero but never reaches it. We used the equation (lim(1/x)[sub]x->infinity[/sub]) - 0 = epsilon as the definition. (Epsilon was our symbol for an infinitesimal).

(Oh, just as a caveat… I’m trying to remember this from 25 years ago so I may not be remembering exactly how the book taught it, this is how it is stuck in my memory. Any errors are probably due to the brain cells the beer killed at graduation.)

It means “reduction”. The standard transliteration, though, is “jabr”, not “gebra”.

AAAAAUUUUUGGG!!!

lim(1/x)[sub]x->infinity[/sub] does not approach zero. it is equal to zero!

You have defined epsilon = 0 - 0.

I thought we’d see your smiling face here. Hi, ultrafilter.

Of course, there are other calculi, such as the Lambda calculus (familliar to those of you with a functional programming background) and the reduction calculus, used in Logic programming.

Didn’t mean to cause you anguish, Dr. M.

I was thinking that lim(1/x)[sub]x->infinity[/sub] was arbitrarily close to zero but never actually got there; hence epsilon was the difference between zero and “arbitrarily close to zero”. Anyway, that’s what I was trying to get across. Perhaps I’m using the wrong terminology for the concept? As I said in my caveat, it has been a not-so arbitrarily long time since I took the class.

And al-Khwarazmi became algorism which became algorithm. So a second major term in Science was due to the same guy that only a few people ever heard of.

At my old college, the Math profs were split over which style of calculus to use. They settled on a book that mainly used limits but had infitesimals in the appendix. The word got out to students to be careful about switching profs during the calc sequence. You could end up with one who used a method different from what you had learned so far. Same book but…

Bryanmcc, your question was specific, so I’ll try to give what I think is a specific answer.

Arabic etymologies aside, most school subjects in the area of number manipulation and calculation are listed in the standard English way we list the various learned disciplines and subdisciplines–sans “the.” We study Astronomy, Chemistry, History, Spanish: not The Astonomy, The Chemistry, etc. (Note that other Romance languages do often include the definite article in such references.)

The typical subjects of “math” (British: “maths”) class are: addition, subtraction, multiplication, division, fractions, algebra, geometry, trignometry, and…The Calculus. The last is allowed to have a definite article because, as stated by another poster, the subject is not “calculus” (on analogy with algebra or trignometry) but rather “study of The Calculus of Infinitesimals,” TCOI remaining its name, more or less, even if the infinitesimal concept has been supplanted by the use of limits. In other words, it has a “the” for the same reason there’s a “the” in The Pythagorean Theorem, The Bill of Rights, The United States of America, etc. (Capitalizing doesn’t matter.) The only reason it seems odd is that the other parallel subjects LACK a “the.”

…And then things get fouled up further by the tendency of school catalogs and class lists to go for the short version, dropping the “the” there, while retaining it (sometimes) in other places.

>> Why is it called “the” calculus?

Why? Because it’s in “the” kidney! That’s why!

Maybe it’s just a local thing, but I took economics in Canada, including two calculus courses and quite a lot of calculus, and I never once heard ANYONE say “the calculus.” Not our professors, not our TAs, nobody.

It really used to bug me when my physics professor would say : “and then, using the calculus, blah blah blah, you get 7 Newtons” somebody told me it was because calculus is a noun, even though it seems like it’d be a verb because it’s something you do. Yes, I know, you don’t calculus. You do calculus. So do is the verb and calculus is the noun? Just when I think I understand it it gets more confusing.