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As far as I remember,

Calculus was invented/discover/founded by Newton.

But what he was trying to achieve that made him find the limit of of difference approaching zero.

how far did he get into calculus ( did he also found the integration? differential equations?)

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  • $\begingroup$ Newton is currently considered to be a co-founder of calculus. Some credit also goes to Liebniz, but some also goes to some Indian mathematicians before Newton and perhaps even Archimedes before them. (This topic is still controversial.) Are you asking about about Newton's purposes or also those of the others I mentioned? $\endgroup$ – Rory Daulton Jul 18 at 0:12
  • $\begingroup$ i am sure many other topics helped in Calculus finding. but my question is about calculus as separate area in mathematics. who ever decided to study that area and put the rules of differentiation such as d/dx x^2 = 2x $\endgroup$ – asmgx Jul 18 at 0:17
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    $\begingroup$ @Rory Daulton, never heard or read of pre-Archimedes Indian mathematicians who discovered calculus. Is there a scholarly reference? There is a recent wave of fictitiously attributing everything to "ancient" Indians on Wikipedia and elsewhere. BBC has already made a report on these type of stories. Most of them are jokes! bbc.com/news/world-asia-india-46778879 $\endgroup$ – M. Farooq Jul 18 at 2:22
  • $\begingroup$ @M.Farooq: Perhaps i wrote it badly, but I mean that Archimedes came before the Indians who came before Newton. I was referring primarily to Madhava of Sangamagrama. I have seen claims that he founded calculus--I do not know enough to take any position on that claim. $\endgroup$ – Rory Daulton Jul 18 at 8:20
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    $\begingroup$ @RoryDaulton, The example of Wiki article Madhava of Sangamagrama itself is a proof of the recent wave of assigning everything to "ancient" India. My biggest question is where are those ancient books which had all the information? Look at the big claims "Discovery of power series expansions of trigonometric sine, cosine and arctangent functions; infinite series summation formulae for π". If everything was invented back in 1350s, then modern mathematicians just wasted their time and re-invented the wheel. $\endgroup$ – M. Farooq Jul 18 at 12:11
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You remember incorrectly. Calculus was found by Archimedes, G. Saint-Vincent, Galileo, Kepler, Descartes, Pascal, Cavalieri, Fermat, Barrow, Wallis, Brounker, Huygens, Leibniz, J. Gregory, N. Mercator, Newton, Cotes, Taylor, Torricelli, Bernoulli brothers, to name only the most famous ones. As every big enterprise, this was a collective enterprise.

The problems which led to its development are: finding areas and volumes (integration), finding tangents to curves (differentiation), finding maxima and minima of functions and functionals (calculus of variations) and expansion of functions into power series which was used for solving differential equations arising in geometry and physics.

But if by "calculus" you only mean the differentiation rules and Newton-Leibniz formula, these were found by Newton and Leibniz, independently. But this is just one theorem of calculus.

To answer your second question, yes, Newton (and Leibniz and Bernoulli) also knew integration and differential equations. Integration was developed by Eudoxus and Archimedes, and this is the oldest part of calculus. Differentiation as a tool of finding extrema was also used by Archimedes (and by Fermat, and by others).

Ref. N. Bourbaki, Elements of the history of mathematics.

Remark. Since my mentioning of Archimedes triggered so many comments, let me cite Nicolas Bourbaki, the essay on History of Calculus (my own translation):

The greatest mathematical discovery of the Greeks was their method of treatment of problems which we call integral calculus. Eudoxus gave the first examples of application of this method when he determined the volumes of a cone and a pyramid; this reached us in more or less adequate description by Euclid (VII, Prop. 7, 10). But most importantly, almost all works of Archimedes are devoted to these problems, because of an exceptional luck we can read them in the originals, in his beautiful Doric dialect..

He also mentions that Archimedes was by far the most cited mathematician in 17th century.

Let me add that all surviving works of Archimedes are easily available in English translation to which I send all those who have any doubts about who invented integration. And many commentaries to them are also available. But for a short and non-technical history of calculus in 17th century (and the role of the Greek heritage in it) I recommend the article of Bourbaki cited above.

BTW, Newton himself described his main contribution to calculus as:

One can solve any differential equation by plugging a power series with indetermined coefficients to it, and find the coefficients one-by one.

(I slightly modernized his language). This is not taught in modern elementary courses.

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  • $\begingroup$ does that mean Archimedes used integration of 2x = x^2 + c ? $\endgroup$ – asmgx Jul 18 at 4:46
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    $\begingroup$ I don't believe that, the ancient Greek math was far below this level. I think the Newton & Leibnitz part is the truth. Archimedes might have inventend some algorithm which could be retroactively interpreted as some application of calculus, but I think it was not. $\endgroup$ – peterh Jul 18 at 5:26
  • $\begingroup$ @peterh I'd say you're in the minority there - most analyses I've read conclude the discovered palimpsests clearly show use of differential elements. $\endgroup$ – Carl Witthoft Jul 18 at 13:19
  • $\begingroup$ @CarlWitthoft Did they really use the concept of the infinitesimals? As far I know, one of them made a practical experiment, to calculate the volume of the sphere with sand; what is quite unusual from their mentality. I think it only shows, they had really no idea and it looks for me like a last resort. $\endgroup$ – peterh Jul 18 at 14:05
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    $\begingroup$ Archimedes's works are indeed easily available in English, and there is no integration or calculus in them. Bourbaki's Elements is not a serious source on history, what they wrote is a historically themed introduction into modern mathematics. $\endgroup$ – Conifold Jul 18 at 20:06

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