# Why was calculus invented?

Can anyone show me any flaw for which there was a need of inventing calculus or give me an example where normal maths fail and we need calculus?

• To compute the are of a "complex" shape; see Quadrature. See History of the Calculus. – Mauro ALLEGRANZA Jul 6 '17 at 8:03
• Limits provide concise approximations to expressions and formulae which would otherwise be computationally intractable. The central limit theorem, the prime number theorem, the Poisson limit theorem, etc. – Thoth Jul 6 '17 at 8:13
• Newton needed derivatives to handle motion of bodies. – md2perpe Jul 6 '17 at 10:49
• @Thoth, don't confuse limits with the calculus. Limits are an important technical tool but the main ideas of the calculus were formulated in a language that mostly avoided limits, though Newton's "ultimate" terminology was equivalent to them. – Mikhail Katz Jul 6 '17 at 11:34
• @MikhailKatz: limits do belong to calculus. – Yves Daoust Jul 7 '17 at 7:54

## 5 Answers

This is actually an excellent question and at the time calculus first emerged in the 17th century in the work of Leibniz and Newton (in alphabetical order, without implying any priority), many top-level scholars were sceptical and in fact issued a number of challenges. For example, Huygens adopted the position that the problems Leibniz can solve with the new calculus are no different from the problems one can solve with the old methods, and the difference is only in the language. Huygens may have been at least partly right if he was referring to earlier techniques such as Fermat's adequality which was indeed a powerful method that could solve many problems that today we typically think of as standard ware of the calculus: finding tangents to curves, maxima and minima, areas, centers of mass, etc. In fact, Lagrange considered Fermat to be the inventor of the calculus.

Half a century later, Bishop Berkeley was still vituperating against the new calculus, but the tide has clearly changed and the Fermat-Leibniz-Newton framework was an undeniable success in the hands of Euler.

Something similar is happening today, with Abraham Robinson's "legalisation" of infinitesimals in his 1966 book. Half a century later today, opposing voices are still being heard, but the tide is clearly changing, with high-profile advocates like Fields medalist Terry Tao publishing applications and new insights based on Robinson's framework on an almost monthly basis.

Appreciating the advantage of the calculus over these earlier techniques involves detailed knowledge of the calculus and it is not that easy to give an elementary problem that can be solved by the calculus but not by Fermat's adequality, for example. Using the calculus does simplify calculations significantly compared to the earlier methods.

One specific example that could be mentioned is finding the general tangent line of the cycloid curve. Fermat was able to do it by means of an intricate geometric argument relying on his technique of adequality; see this publication for details. Modern proofs using calculus are more straightforward.

• What is your example problem ? – Yves Daoust Jul 7 '17 at 8:43
• @YvesDaoust, as I pointed out in my answer it is not that easy to give such an example, because of a fairly advanced state of the field even prior to the invention of the calculus. More advanced problems related to tangents to curves, maxima and minima, centers of mass, areas, also variational problems would challenge the earlier techniques and demonstrate the superiority of the new calculus, as I suggest in my answer. Basically my point is that elementary problems don't demonstate the superiority of the calculus. – Mikhail Katz Jul 7 '17 at 8:48
• Try to prune this essay to provide a concise answer to the question posed by the OP. – Yves Daoust Jul 7 '17 at 9:18
• @YvesDaoust, which parts should I delete? – Mikhail Katz Jul 7 '17 at 9:24
• It is not that easy to tell you, because as you know, I am not very versed in the history of Mathematics. – Yves Daoust Jul 7 '17 at 12:51

Basically, it was created to compute areas of regions enclosed by curves and to determine tangent and normal lines to curves.

Invention of Calculus was in order to determine area under a curve and study the rate of change.

What is the speed of a dropped stone after 1 second ?

• This was known to Galileo I believe several decades before Newton introduced the calculus. – Mikhail Katz Jul 7 '17 at 7:59
• @MikhailKatz: this is calculus though. You can't address this problem without a concept of instantaneous speed. Who studied it in the first place is irrelevant. – Yves Daoust Jul 7 '17 at 8:05
• I think that's a rather ahistorical remark, Yves. It is only today that we automatically assign such problems related to parabolas to the domain of calculus. Actually when I was in highschool this problem was solved without resorting to calculus because we hadn't learned it yet. Motion under constant gravity I think is a counterexample to the necessity of calculus to solve concrete problems, and only reinforces the OP's question rather than answering it. – Mikhail Katz Jul 7 '17 at 8:13
• @MikhailKatz: the question is ahistorical. It is requesting a sample problem. It is about why, not who nor when. – Yves Daoust Jul 7 '17 at 8:15
• @YvesDaoust, I disagree with your comment. The question clearly has a historical component related to the "invention of the calculus". – Mikhail Katz Jul 7 '17 at 8:41

Rectification of the ellipse (computation of the perimeter length) requires a special function known as the Elliptic Integral, which can only be defined in the frame of calculus.

https://en.wikipedia.org/wiki/Ellipse#Circumference

• Calculus was not invented for computing the arclength of ellipses. – franz lemmermeyer Jul 9 '17 at 12:03
• @franzlemmermeyer: didn't you read the question ? "give me an example where normal maths fail and we need calculus" – Yves Daoust Jul 10 '17 at 7:53
• I can read "Why was calculus invented?" Can you? – franz lemmermeyer Jul 10 '17 at 10:12
• @franzlemmermeyer: +1 for the bad faith. – Yves Daoust Jul 10 '17 at 10:14