Talk:Calculus/Archive 5

This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

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Archive 3

Enough already

Most of this page now consists of arguing whether calculus should be defined in terms of limits or not. This talk page is for a discussion about the article "Calculus", not the subject of calculus itself. Until edits are made, I suggest that, as per Arcfrk's suggestion of July 23 that we get back to making the article better rather than discussing this topic which, as far as Wikipedia policy would indicate, should not be done on this page. Whether calculus should or should not be defined in terms of limits is immaterial. The overwhelming majority of calculus texts are oriented that way, and until an alternative system gained popularity (that is, a similar number of published works have a different framework) we should not introduce fringe opinions.

I'm not going to contribute any more to a discussion of calculus itself: it's the article that matters. Make the edits; then we'll address those. Xantharius 23:06, 6 August 2007 (UTC)

Oh please, no one is going to make edits only to have these removed. This is your turf and you are the king. Long live Wiki sysops and administrators! 76.31.201.0 00:10, 7 August 2007 (UTC)

This isn't about sysops... this is about wikipedia policy. Most of the people you have been conversing with are not admins and your mention of them is not relevant the situation.--Cronholm¹⁴⁴ 00:22, 7 August 2007 (UTC)

Seconded, I plan on archiving this whole little debacle in two days. If anyone wishes to continue this conversation, either take it up on a talkpage or I can create a sandbox.--Cronholm¹⁴⁴ 23:37, 6 August 2007 (UTC)

An observation

the 'applications' section uses the word calculus too much. Am I crazy?

Quite possibly, the applications section was created in a piecemeal fashion so it doesn't have a coherent theme. I will try to streamline it.(remember to sign your posts!)—Cronholm¹⁴⁴ 11:02, 15 August 2007 (UTC)

Polynomial calculations

Some kind-hearted anonymous editor (not me) wants to add a computation of the derivative of xⁿ to the article. Rather than reverting back and forth, perhaps we could discuss it here?

I for one am opposed to adding the computation. I think it is a little to specific and concrete for an overview article like this. However, I am willing to listen to other views. 141.211.62.20 13:53, 15 August 2007 (UTC)

I agree. For this article, the computation of the derivative of

f(x)=x^{2}

at a particular point is enough to demonstrate the technique. The article derivative might be a place for derivation of power rule, product rule, quotient rule, and chain rule. Rick Norwood 14:30, 15 August 2007 (UTC)

I think it should be included in the article since the power rule is a very fundamental computation for determining the derivative. Besides, the difference quotient is included, so why not the power rule?Dannery4 02:47, 19 August 2007 (UTC)

I think this article is already long enough. There's a "see also" link to calculus with polynomials, which covers the application of the chain rule that Dannery4 is interested in. Isn't that enough? DavidCBryant 10:47, 19 August 2007 (UTC)

Perhaps you all are right. I withdraw my opinion. As a side point though, I was commenting on the power rule, which is different than the chain rule.Dannery4 20:41, 19 August 2007 (UTC)

The difference quotient is included because it's more basic; it's the definition of a derivative. The computation for

f(x)=x^{n}

is almost exactly the same as the one for

f(x)=x^{2}

and hence it adds little to the article. Thus, it should not be included.

There was a small discussion on it a month ago; see Talk:Calculus/Archive 4#power rule for derivatives?. -- Jitse Niesen (talk) 13:57, 19 August 2007 (UTC)

The so-called "power rule" is just a special case of the product rule. (Did I say chain rule? Oops! I should have said "product rule".). In other words, once I know the product rule, then the "power rule" is easily derived by induction on the exponent (i.e., (x²)' = x + x = 2x, etc). I guess you can say it's "different", but to me the two rules look like more or less the same thing. DavidCBryant 22:30, 19 August 2007 (UTC)

The Calculus

Isn't calculus often called "The Calculus"? Zginder 23:43, 7 September 2007 (UTC)

Not as far as I know. —METS501 (talk) 03:03, 8 September 2007 (UTC)

Actually, you may be right. See for example this page which refers to it as "the calculus". —METS501 (talk) 03:04, 8 September 2007 (UTC)

This is an older usage, not common today. Rick Norwood 13:03, 8 September 2007 (UTC)

Maths professors sometimes still refer to it as the calculus. Zginder 13:36, 8 September 2007 (UTC)

The full name is "The calculus of infinitesimals" (cf "The calculus of variations"). I suppose one could drop "infinitesimals" and keep the definite article, but in itself, it does not make sense. I think it would be a good idea to discuss the etymology of the term, perhaps, giving some time scale of what was the subject called in different times. Arcfrk 22:01, 8 September 2007 (UTC)

This issue has already been discussed in archive 2 (here), but Arcfrk's suggestion would fix the problem for posterity (ie people asking the "The" question once in a blue moon). So I guess the question becomes... who would like to trace/(find someone who already has traced) Calculus' etymology from Newton and Leibniz to the present day? Then put it in an article (Calculus or History of calculus)... Any takers? —Cronholm¹⁴⁴ 08:27, 10 September 2007 (UTC)

Perhaps the choice of "Calculus or "The Calculus", in modern usage, is largely dictated by the speaker's or writer's nationality? I know in the US we tend toward informality in speech, which would lead to dropping the "The", I think. We also aren't big on Mr. and Mrs. these days, and workers often address their employers by their first names. I think in some other countries this is very different, with much greater emphasis being placed on formality and status-based rules of propriety, and greater respect being given to, well, greater respect. Many of us here are somewhat uncomfortable being addressed with a title and our last name, given the prominence in our culture of ideas of equality and populism, preferring a simple first name form of address, and perhaps we unconciously believe that branches of learning such as Calculus would experience similar discomfort if burdened with the distinction of a formal title! 68.46.96.38 10:19, 16 September 2007 (UTC)

Yes, I'm a latecomer to this discussion, but I did (believeitornot) come to this page looking for an explanation of "the calculus". I think a note in the main article would be most helpful. (I'd add it, but I am clearly not qualified. :) ) CSWarren 19:30, 12 October 2007 (UTC)

BCE and CE versus BC and AD

Someone changed all of the BCs and ADs to BCEs and CEs, respectively, and then they were reverted. The "common era" versions are, as far as I was aware, the less objectionable. Why were they changed, and is there a Wikipedia policy on this? It's not a religious article referencing Christianity: is there a good reason for having BC and AD? Xantharius 03:43, 14 September 2007 (UTC)

The policy accepts both but discourages reverts back and forth; this issue has come up before(look at the history and archives), and I don't care as long as we just stick with one. I believe that BC and AD were the original notation used on the article, so I would like to leave it at that to discourage rewarding wholesale changes of this variety. —Cronholm¹⁴⁴ 05:28, 14 September 2007 (UTC)

As I've mentioned before, there are people who go through Wiki changing BC to BCE and other people who go through Wiki changing BCE to BC. There is no way to stop them, so I think it is best to just ignore them. It keeps them off the streets. Rick Norwood 13:21, 14 September 2007 (UTC)

Can a clear definition be derived for Calculus?

This would be integral to improving the quality of this article. It should be of the form: Calculus is the branch (or field?) of mathematics which does such-and-such actions. The following is perhaps quite flawed, as I am not really knowledgeable about calculus, but perhaps it could be used as a starting point, and cleaned up?: "Calculus is the branch of mathematics which deals with the behavior of numerical values as they change." For a definition to be useful, it should be possible, using the definition, to differentiate if any specific thing is part of Calculus, or if it is not. If a clear definition of the term "Calculus" can't be achieved, can we really say "Calculus" is a meaningful term? 68.46.96.38 10:55, 16 September 2007 (UTC)

The problem is that "Calculus" is really a college course rather than a branch of mathematics. Many calculus textbooks say something like what you say, "Calculus is the mathematics of change," and I have no objection to using that definition in the article.

In fact, what is called "calculus" in undergraduate education is called "analysis" in graduate mathematics, and is one of the three major branches of abstract mathematics that follow foundations: topology, abstract algebra, and analysis. These three branches are, roughly, the study of point sets, the study of number sets, and the study of functions. "Calculus" is the undergraduate analysis course that studies functions but sweeps some of the more difficult problems under the rug.

I've done a minor rewrite of the intro to attempt to make this clear. Rick Norwood 14:03, 16 September 2007 (UTC)

In my opinion, calculus is clearly not a branch of mathematics, it is simply a bunch of rules for calculating integrals and derivatives (and some sequences). Looking up the word calculus on wiktionary gives "Any formal system in which symbolic expressions are manipulated according to fixed rules" (as the general meaning of the word) and as an example it lists "vector calculus", and it is really in this sense that "the" calculus should considered. I guess what I'm trying to say is that the first line should probably be something more like "In mathematics, calculus is a set of rules for computing derivatives, integrals and certain other limits, and constitutes..."; though this probably isn't great either. But I definitely think that it shouldn't be called a branch of mathematics. RobHar (talk) 10:01, 26 November 2007 (UTC)

A word means what the people who commonly use the word intend it to mean. The current meaning of calculus is that branch of mathematics which considers limits, derivatives, integrals, and infinite series. While some teachers only teach rules (this is called "cookbook calculus") most prove theorems, though not as many theorems as we once did. All major calculus books include theorems and proofs. Vector calculus is not about, for example, dot and cross product -- that's Linear Algebra -- but about integrals and derivatives of vector valued functions. Rick Norwood (talk) 14:25, 26 November 2007 (UTC)

What would you say is the difference between Calculus and Analysis? --Lambiam 15:34, 26 November 2007 (UTC)

The branch of mathematics that considers limits, derivatives, etc. is analysis. The set of methods used to compute these things when considering a very specific class of functions of one real variable is what calculus is (one could include multivariable calculus and vector calculus). There was probably a point in time when calculus was a branch of mathematics, when people were proving new things and developing the methods, but it's probably been a while since there's been an article written on calculus or an NSF grant given out for calculus research. Similarly, there was a point when the theory of determinants could probably have been considered as a branch of mathematics, but math evolved and this became a specific set of results in the branch algebra called linear (or multilinear) algebra. Though it is true that some theorems are proved in calculus classes, most of the time this is handwaving, and calculus assignments basically never include questions about proving a theorem, and are definitely focused on seeing if the students are able to apply the rules taught to them. RobHar (talk) 20:11, 26 November 2007 (UTC)

You and Rick are basically saying the same thing. RobHar, I agree with what you are saying to a point, but calculus is a subset of analysis, just the introduction to it basically. And the amount of proofs involved in a calculus class varies widely from none to a fairly sophisticated level, depending on the institution and level (regular, honors, etc). I guess I have to agree that analysis would be the proper branch of mathematics and calling calculus that is stretching it a bit. I think it's possible to meet in the middle though. Perhaps, simply moving the sentence "In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions." to the end of the first sentence of the article would cover it. That way the reader sees that analysis is the broader concept right away. Add to that a way to make it clear analysis is the branch and calculus is a subset of/introduction to it and that should encompass what both of you are saying. - Taxman ^Talk 00:09, 27 November 2007 (UTC)

Hey Taxman, thanks for weighing in. I guess I just don't quite view it that way. For example, many (most?) places have calculus courses and analysis courses with pretty much the same course description (ie the subject matter covered overlaps, but the point of view is different), and if you look at places like Harvard or Princeton, many (most?) students that are going into math don't even take calculus and instead go straight to analysis, because knowing calculus isn't necessary to understanding analysis. I do view calculus as a subset of analysis, but I do not view it as a logical beginning (in terms of subject development, that is) of the subject, but rather a first example of concrete calculations. Like how modular arithmetic is a good introduction to rings and ideal, so one might begin an abstract algebra course by discussing Z/n and doing many calculations and theorems, but one could have simply started by defining rings and ideals. I guess the one thing I'm not sure about is that it seems calculus courses in Europe are more rigorous than ones in North America, perhaps we are having a disagreement that is based on regional differences. I guess, in my experience taking many calculus and analysis classes, and in teaching, the difference between regular and honours calculus classes has mainly been the difficulty of the problems, perhaps a bit the level of rigour, but proofs were never expected to have been understood for exams. RobHar (talk) 03:07, 27 November 2007 (UTC)

I don't know about many places. I can only tell you about the university I went to. The honors calc sequence was pretty theory heavy and proofs were very much demanded, on exams as well. Of course there was some cookbook too it, since it wasn't the purely theoretical class, which was available as well. The regular calc sequence was entirely cookbook. - Taxman ^Talk 14:56, 28 November 2007 (UTC)

Once again, it is not our place to write articles based on our low opinion of the American school system, but rather to report what standard sources say. I appreciate Taxman's effort to mediate, but limiting "calculus" to functions of one (real?) variable won't do. The course I'm teaching now covers functions of several variables and is called "calculus" and taught out of a book called "calculus", as was the course I took as a Freshman at M.I.T., where the textbook was Calculus and Analytic Geometry by Thomas.

Mathematics includes not only areas of current active research but also the discoveries of the past. Calculus does not stop being a branch of mathematics even if it is no longer an area of active research (which the non-standard analists would dispute). Rick Norwood (talk) 20:23, 27 November 2007 (UTC)

Just a note, sorry to insert here. I wasn't the one that limited calculus to one variable. I agree with you on that point Rick, I just didn't happen to specify my disagreement with that point of Rob's. - Taxman ^Talk 14:56, 28 November 2007 (UTC)

Hey Rick Norwood, I don't really think you're addressing my points. To address yours, firstly I do not have a low opinion of the American school system, and my opinion on that subject has nothing to do with what I'm trying to talk about (and I'm also not sure why the sentence begins with "once again"...) . Perhaps your next comment is only about Taxman, but I included above that multivariable calculus and vector calculus (and probably some complex integral techniques) can be considered calculus (and I do consider them calculus). I do believe that mathematics includes calculus (as I mentioned above that it was "a subset of analysis"), but I definitely disagree that calculus is a branch of mathematics. There was probably a point where calculating the area of a disc was a branch of mathematics, or the study of quadratic forms, or as I mentioned above the theory of determinants, but these branches became parts of bigger more general branches. Perhaps the non-standard analysts are studying how to formalize certain manipulations in calculus such as differentials, but this is not calculus, just like the study of the formal system of peano's axioms is not grade school arithmetic.

The points I'd like you to address are the following, and if they are addressed and I'm shown to be wrong, I'll have no problem agreeing with you.

1. You mentioned standard sources for the definition of calculus, which of the many included in the article say calculus is a branch of mathematics?

2. How do you account for the fact that universities (including MIT, your alma mater) have both courses called calculus and courses called analysis that cover the same material? This would lead me to believe that standard sources (namely university math departments) consider these to be different things.

3. Do you think that most calculus courses include a nontrivial amount of exam questions testing the student's understanding of proofs of the theorems, and/or their capability to produce more proofs?

4. Do you believe that the theory of determinants is (still) a branch of mathematics? You address this by saying calculus doesn't become not a branch of math, but you seems to say more generally no branch is ever demoted. Perhaps we're just disagreeing on the term branch.

My main actual disagreement with what Taxman said was a minor one, and involved the fact that a view calculus as a subset, not the beginning of the branch of analysis (in a logical sense, not a historical sense). What I'd like to see at the moment is something meaning "Calculus was an important branch of mathematics for hundreds of years and remains a part of mathematics as a subset of the modern branch of Analysis." (which is, I think, along the lines of what Taxman was suggesting). Thanks. RobHar (talk) 23:17, 27 November 2007 (UTC)

Would it help if we use "a field of mathematics", a formulation also used in Differential calculus, Calculus of variations, Vector calculus, and many other articles? --Lambiam 08:22, 28 November 2007 (UTC)

I have no problem with "field" of mathematics or "area" of mathematics, but I prefer "branch" because it describes the role of increasing specialization in mathematics. The first "branch" is pure vs. applied, then pure branches into geometry, number theory, and analysis, then analysis branches into real and complex analysis, and so on. My own branch (twig?) is knots on the double torus.

So, why not title this article "analysis" and redirect "calculus"? Because calculus is the more common word. Articles like this are of value to laypersons, not to mathematicians. To a layperson, the first thing they need to know about any abstract knowledge is: into which major category does it fall (mathematics), what are the prerequisites for learning it (algebra, geometry, and analytic geometry), and what are its applications (advanced math, science, engineering, computer science).

Now, to RobHar's four questions. 1) The first reference I happened to pick, The Concise Columbia Encyclopedia, says, "Calculus, branch of MATHEMATICS that studies continuously changing quantities." 2) Calculus and analysis do not cover the "same" material. Analysis builds on calculus. From Royden, "Real Analysis", "It is assumed that the reader already has some acquaintance with the principal theorems on continuous functions..." 3) As I've said before, what calculus is does not depend on how well or badly it is taught. But, to answer your question, no, most calculus teachers do not ask for theory on exams. Most give cookbook exams because otherwise most students would flunk. M.I.T. is not my alma mater. I flunked out, as did about half of the students in my day. Today, admission to M.I.T. is so highly prised that the effort to retain students has resulted in lowered standards. As of a few years back, M.I.T. started offering courses in "developmental math" ("developmental" is math ed jargon for "remedial"). But "math ed", a very important area in education, is not the same as "math". 4) I would say that one of the branches of abstract algebra is vector spaces, that at an elementary level, the study of vector spaces are usually called linear algebra, and that the study of determinants is a branch of linear algebra, and that the study of SL(2) is a branch of the study of determinants. Of course, the study of SL(2) is also a branch of group theory, and here the tree analogy breaks down, since in some sense all mathematics is one. But to the layperson, the picture of a branching subject is useful, and corresponds roughly to the list of prerequisites for undergraduate math courses.

I do understand your point -- that the way calculus is taught today in the US it is more a set of rules and not really mathematics at all. This is a valid criticism. But the only important point in all this is that standard sources describe calculus as a branch of mathematics. We should too. Rick Norwood (talk) 14:09, 28 November 2007 (UTC)

I guess after thinking and reading more, I have to say the word branch is misleading. It's not a proper branch anymore. I agree field is better and covers the important part of what the word branch is used for anyway. It's unfortunate that some sources use it that way, but using that word just because they use it in their introduction isn't an entirely proper way to use sources anyway. Unless the central point of the source is to describe whether calculus is a proper branch or not, then the choice of wording used in the source isn't definitive. More specifically to what the intro should say, I'm not opposed to what Rob has said "Calculus was an important branch of mathematics for hundreds of years and remains a part of mathematics as a subset of the modern branch of Analysis." being part of the intro, but it tells what it was not what it is. It is currently an important part of math education (and nearly all sciences by extension) and it is a field of mathematics, a subset of analysis. I think we need to get that accross first. - Taxman ^Talk 14:56, 28 November 2007 (UTC)

Another "standard" encyclopedic sources, the 1911 Brittanica, as rendered by LoveToKnow:[1]

INFINITESIMAL CALCULUS. 1. The infinitesimal calculus is the body of rules and processes by means of which continuously varying magnitudes are dealt with in mathematical analysis.

Personally I'd say that Calculus is a tool, comprising a collection of notations and methods initially invented independently by Leibniz and Newton, which is very useful in tackling some, but not all, problems studied in Analysis. Consider the function f defined by f(x) = 0 for irrational x and f(p/q) = q⁻² for rational arguments given in simplest terms. When we prove that this function is continuous and has a vanishing derivative on the irrationals, we don't use any of the methods of the Calculus, which are useless here. Yet this is clearly the province of Analysis. On the other hand, when we apply the chain rule, or do integration by parts, we are wielding the tools of Calculus. --Lambiam 16:19, 28 November 2007 (UTC)

I agree with what Taxman and Lambiam have just said. On Taxman's comments, indeed my suggestion "Calculus was a branch..." isn't as good as it could be and I don't think it should be the first sentence, I just think the idea of what it says should appear in the intro somewhere. I also agree with Rick Norwood's response to my 4th question, as determinants are a branch of linear algebra, one could probably say "Calculus is a branch of analysis" (hey perhaps the article could start "In mathematics, calculus is a branch of analysis" and go on to say that for over 150(?) years it was its own branch of mathematics).

To respond to Rick Norwood's other points, firstly Royden's book, in the words of the publisher is the "classic introductory graduate text", so yes it probably assumes prior knowledge of theorems on continuous functions. Secondly, my point is not that calculus is poorly taught, I in fact think that calculus is closer to what Lambiam above has said. To support this, consider the fact that calculus was indeed invented in the late 1600's and was only formalised in the 1800's (by Weierstrass, Cauchy, Riemann, etc.) and that during the period between these times, calculus was indeed a set of rules (L'Hospital's Rule, the chain rule) and methods (Newton's book Method of Fluxions) to solve certain problems. The proofs given when one covers calculus theoretically in an undergraduate course are from the 1800s, for example, one uses the Riemann integral, which did not exist before the 1840s. I may go so far as to say that, yes, certain calculus courses incorporate some proofs, but that these proofs are from analysis, not calculus, but this isn't we need to discuss right now, nor put in this article.

In summary, I think something like "In mathematics, calculus is a branch of analysis" or "In mathematics, calculus is a field of analysis" is more appropriate. I also think that it should be said that calculus was a branch of mathematics in its own right for a long time, and that it then became a subset of analysis. RobHar (talk) 01:12, 29 November 2007 (UTC)

New stuff about Archimedes and Calculus

http://www.sciencenews.org/articles/20071006/mathtrek.asp Gwen Gale 09:17, 8 October 2007 (UTC)

Summary: Heretofore unknown texts --> Early conception of limiting values by Archimedes used in calculations (parabola triangle example)+ distinction between "actual" infinity vs. "potential" infinity for calculations. Archimedes worked with both. The former only in his lost work.

—Cronholm¹⁴⁴ 09:40, 8 October 2007 (UTC)

Excellent article. There should definitely be a sentence about it here, and a paragraph in History of Calculus. Do you want to write it, or shall I? Rick Norwood 13:01, 8 October 2007 (UTC)

You go ahead, I have a term paper to write. :) —Cronholm¹⁴⁴ 03:03, 9 October 2007 (UTC)

Calculus of Antiquity

Ideas of calculus were developed earlier, in Egypt...

Sorry I had to delete Egypt. There has NEVER been any evidence that the Egyptians had developed any for of calculus whatsoever. We do however have the Moscow Papyrus which shows that the Egyptians had correct calculations for complicated volumes such as the frustum of a pyramid, but NEVER have we been given any evidence of them developing calculus.

I hope others can follow suit and check the claims made to other parts of the world. And when claims have been verified please find appropriate dates for the developments and cite resources. --123.100.92.83 19:18, 19 October 2007 (UTC)

There are many such claims, and they are hard to check. We really need someone who reads the language to check such claims, but how many Wikipedians read Sanscrit? Rick Norwood 12:51, 21 October 2007 (UTC)

The pump don't work 'cause the vandals took the handle.

Good work fighting vandalism, Gscshoyru. Rick Norwood 12:53, 21 October 2007 (UTC)

History

The History section is a bit long. I don't think that it is worth mentioning every time that the area of a circle was determined. The fact that Cauchy and Riemann only get a passing mention, and 3 different people are mentioned for calculating a circle's area at different points in history is sort of rediculous. I would suggest that the history section be trimmed down, to make mention of the fact that similar concepts had been developed before Newton and Leibniz.(Lucas(CA) (talk) 06:09, 16 December 2007 (UTC))

Calculus was not developed in India. At the very least, just find any Calculus textbook, go to the index, and look up the name Aryabhatta or any of the names I am removing. Their names are not in any index. And for good reason: they did not develop calculus. Aryabhata's contributions belong in the geometry article. Calculus relates areas of functions to their antiderivatives. Aryabhatta nowhere relates the idea of antiderivative (which had not been developed by his time) to finding the area under a curve. The method of exhaustion is a geometric technique, it is not calculus. Look at the first and second "fundamental theorems of calculus" for proof of this. They relate integrals to their antiderivatives. This connection was not known until the 17th century.

Every calculus text on the market today mentions nothing about Aryabhata (and they go to lengths to list the names who made important contributions to the development of calculus. Names like Wallis, Lagrange, Rolle, de Fermat etc that were mentioned in the previous wiki pages.) This goes for the textbooks by Larson/Hostetler, Stewart, and Thomas'.

Furthermore, look at the work cited to support this person's claim. Do a control-f search on that page for 'calculus'. Nothing comes up, again because Aryabhatta did not contribute to this field. So why is he mentioned in a page about calculus? That is why I am removing his name and picture. If someone wants, they can add him to the page about geometry or algebra, this page is about calculus. These distinctions are important if we want to keep things organized. —Preceding unsigned comment added by 70.185.199.182 (talk) 14:30, 9 May 2008 (UTC)

You have removed an entire paragraph of non-Western contributions. I agree with your assessment that Aryabhata should be removed, you also deleted references to Indian and Islamic mathematicians who are relevant to the development of calculus. For instance, the Kerala school is widely recognized as having made important contributions to the ideas of calculus in non-Western civilization. silly rabbit (talk) 15:40, 9 May 2008 (UTC)